Generally, there room two types of logarithmic equations. Study each case carefully prior to you start looking at the worked examples below.

You are watching: How to solve logarithms with different bases

Types of Logarithmic Equations

The first type looks like this.
*

If you have actually a solitary logarithm on each side of the equation having actually the very same base climate you can collection the debates equal to every other and solve. The arguments here room the algebraic expressions represented by colorblueM and colorredN.


*

If you have a solitary logarithm on one side the the equation climate you have the right to express it together an exponential equation and solve.

Let’s learn just how to fix logarithmic equations by going end someexamples.

Examples of exactly how to deal with Logarithmic Equations



Since we want to change the left side into a single logarithmic equation, then we need to use the Product preeminence in reverse to condense it. Below is the ascendancy just in case you forgot.


*


Just a big caution. ALWAYS check your resolved values through the initial logarithmic equation.

Remember:

It is OKAY to have values of x such as positive, 0, and an unfavorable numbers.However, it is no ALLOWED to have a logarithm that a an adverse number or a logarithm the zero, 0, when substituted or evaluated right into the initial logarithm equation.

⚠︎ CAUTION: The logarithm that a an adverse number, and the logarithm the zero space both no defined.

log _bleft( mnegative,,number ight) = mundefined

log _bleft( 0 ight) = mundefined

Now, let’s inspect our answer if x = 7 is a precious solution. Substitute back into the initial logarithmic equation and verify if it returns a true statement.


Yes! since x = 7 checks, we have actually a systems at colorbluex = 7.

Example 2: solve the logarithmic equation.


Start by condensing the log expressions ~ above the left right into a single logarithm making use of the Product Rule. What we desire is to have a single log expression on each side the the equation. Be ready though to settle for a quadratic equation due to the fact that x will have actually a power of 2.

Given
Simplify: left( x ight)left( x - 2 ight) = x^2 - 2x
Drop the logs, set the disagreements (stuff inside the parenthesis) equal to every other
Set each variable equal to zero then fix for x.

x - 5 = 0 means that x = 5

x + 2 = 0 means that x = - 2

So the possible solutions arex = 5 andx = - 2. Remember to constantly substitute the possible solutions earlier to the original log equation.

Let’s inspect our potential answer x = 5 and x = - 2 if they will certainly be precious solutions. Substitute ago into the initial logarithmic equation and verify if it returns a true statement.


After check our values of x, we found that x = 5 is certainly a solution. However, x =-2 generates some negative numbers within the parenthesis ( log in of zero and negative numbers space undefined) that renders us eliminate x =-2 as component of ours solution.

Therefore, the last solution is just colorbluex=5. We ignore x=-2 because it is one extraneous solution.

Example 3: solve the logarithmic equation.


This is an amazing problem. What we have actually here are distinctions of logarithmic expressions on both political parties of the equation. Simplify or condense the logs in both sides by using the Quotient dominance which looks like this.


Given
I think we are all set to collection each debate equal to every other because we space able to alleviate the difficulty to have actually a solitary log expression on every side that the equation.
It looks like this after acquiring its cross Product.
Simplify both sides by the Distributive Property. At this point, us realize the it is just a Quadratic Equation. No big deal then. Move everything to one side, and that forces one side of the equation come be same to zero.
This is conveniently factorable. Now set each aspect to zero and solve for x.
So, these space our possible answers.

I will leave it come you to inspect our potential answers ago into the original log equation. You have to verify the colorbluex=8 is the only solution, while x =-3 is not because it generates a script wherein we room trying to gain the logarithm the a an unfavorable number. No good!

Example 4: fix the logarithmic equation.


If you watch “log” without an explicit or written base, it is assumed to have a base of 10. In fact, logarithm v base 10 is recognized as the common logarithm.

What we require is to condense or compress both political parties of the equation into a single log expression. On the left side, we see a distinction of logs which method we apply the Quotient dominion while the best side requires Product Rule because they’re the sum of logs.

There’s just one point that you have to pay fist to the left side. Do you watch that coefficient Large1 over 2,?

Well, we have to carry it up together an exponent making use of the Power dominion in reverse.


Given
Bring up the coefficient large1 over 2 together an exponent (refer come the outward term)
Simplify the exponent (still introduce to the leftmost term)
Then, condense the logs on both political parties of the equation.Use the Quotient dominion on the left and also Product dominance on the right.
Here, i used different colors to show that since we have actually the same base (if no explicitly shown it is assumed to be basic 10), it’s okay to set them same to each other.
Dropping the logs and also just equating the debates inside the parenthesis.
Set each variable equal to zero and also solve for x.

It’s time to examine your potential answers. Once you examine x=0 earlier into the original logarithmic equation, you’ll end up having an expression that entails getting the logarithm the zero i m sorry is undefined, meaning – no good! So, us should ignore or fall colorredx=0 as a solution.

Checking Largex = 3 over 4, confirms that indeed Largecolorbluex = 3 over 4 is the only solution.

Example 5: fix the logarithmic equation.


This trouble involves the usage of the prize ln instead of log to average logarithm.

Think the ln together a special sort of logarithm utilizing base e where e approx 2.71828.


Given
Use Product ascendancy on the appropriate side
Simplify the two binomials by multiply them together.
At this point, I just color-coded the expression within the parenthesis to indicate that us are ready to collection them same to each other.
Yep! This is wherein we say the the stuff inside the left parenthesis equates to the stuff inside the appropriate parenthesis.

Don’t forget the pmsymbol.


Simplifying further, us should obtain these feasible answers.

Check if the potential answers found over are feasible answers through substituting them ago to the initial logarithmic equations.

You have to be convinced that the only valid solution is largecolorbluex = 1 over 2 which makes largecolorredx = -1 over 2 one extraneous answer.

Example 6: fix the logarithmic equation.


There is just one logarithmic expression in this equation. We take into consideration this as the second case wherein us have


We will certainly transform the equation from the logarithmic kind to exponential form, then resolve it.

Given
I color-coded the components of the logarithmic equation to present where they go once converted right into exponential form.
The blue expression remains at its existing location, however the red number i do not care the exponent the the base of the logarithm i m sorry is 3.
Simplify the appropriate side, 3^4 = 81.

You have to verify the the value colorbluex=12 is indeed the equipment to the logarithmic equation.

Example 7: fix the logarithmic equation.


Collect every the logarithmic expressions on one side of the equation (keep the on the left) and also move the consistent to the best side. Usage the Quotient dominion to refer the distinction of logs together fractions within the parenthesis the the logarithm.

Given
Move every the logarithmic expression to the left the the equation, and the continuous to the right.
Use the Quotient dominance to condensation the log expressions on the left side.
Get ready to create the logarithmic equation into its exponential form.
The blue expression stays in its existing location, yet the red continuous turns the end to it is in the exponent that the basic of the log.
Simplify the appropriate side of the equation since 5^colorred1=5.
This is a reasonable Equation due to the existence of variables in the numerator and also denominator.

I would deal with this equation utilizing the cross Product Rule. However I need to express an initial the best side the the equation v the explicit denominator that 1. That is, 5 = large5 over 1


Perform the Cross-Multiplication and then settle the resulting linear equation.

When you examine x=1 earlier to the initial equation, you have to agree the largecolorbluex=1 is the solution to the log equation.

Example 8: fix the logarithmic equation.


This trouble is very comparable to #7. Let’s gather every the logarithmic expressions to the left while maintaining the continuous on the best side. Because we have the distinction of logs, we will utilize the Quotient Rule.

Given
Move the log in expressions to the left side, and keep the constant to the right.
Apply the Quotient Rule due to the fact that they are the distinction of logs.
I used various colors here to show where they go after rewriting in exponential form.
Notice the the expression inside the parenthesis continues to be on its current location, when the colorred5 becomes the exponent of the base.
To settle this reasonable Equation, apply the overcome Product Rule.
Move everything to the left side and make the right side simply zero.

Factor out the trinomial. Collection each element equal come zero then fix for x.


When you fix for x, you should acquire these values of x as potential solutions.

Make certain that you inspect the potential answers native the original logarithmic equation.

You need to agree that colorbluex=-32 is the only solution. That renders colorredx=4 one extraneous solution, so overlook it.

Example 9: deal with the logarithmic equation


I hope you’re gaining the key idea currently on how to approach this form of problem. Here we see three log expressions and also a constant. Let’s separate the log in expressions and also the consistent on opposite sides of the equation.

Let’s keep the log expressions top top the left side while the consistent on the right side.
Start by condensing the log expressions utilizing the Product preeminence to deal with the amount of logs.
Then further condense the log expressions making use of the Quotient ascendancy to attend to the distinction of logs.
At this point, i used various colors to highlight that I’m all set to to express the log equation into its exponential equation form.
Keep the expression within the group symbol (blue) in the same ar while make the constant colorred1 on the appropriate side as the exponent of the basic 7.
Solve this rational Equation using Cross Product. Refer 7 as large7 over 1.
Cross multiply.
Move all terms on the left side of the equation. Factor out the trinomial. Next, set each element equal come zero and solve for x.
These are your potential answers. Constantly check your values.

It’s apparent that as soon as we plugin x=-8 back into the original equation, it outcomes in a logarithm v a an adverse number. Therefore, girlfriend exclude colorredx=-8 as part of your solution.

Thus, the only solution is colorbluex=11.

Example 10: resolve the logarithmic equation.


Keep the log expression ~ above the left, and also move every the constants top top the ideal side.
Simplify.
I think we’re all set to transform this log equation into the exponential equation.
The expression inside the parenthesis continues to be in its existing location while the continuous 3 becomes the exponent the the log base 3.

Check this separate lesson if you require a refresher on how to deal with different species of Radical Equations.


To get rid of the radical price on the left side, square both political parties of the equation.
After squaring both sides, it looks choose we have a straight equation. Just solve it as usual.

Check her potential answer earlier into the initial equation.

See more: What Is A Product Of Primes, How Can We Write 13 As A Product Of Prime Numbers

After law so, you have to be persuaded that without doubt colorbluex=-104 is avalid solution.

Practice through Worksheets


Download version 1
Download version 2

You might additionally be interested in:

Condensing Logarithms

Expanding Logarithms

Logarithm Explained

Logarithm Rules


ABOUTAbout MeSitemapContact MePrivacy PolicyCookie PolicyTerms the Service
MATH SUBJECTSIntroductory AlgebraIntermediate AlgebraAdvanced AlgebraAlgebra native ProblemsGeometryIntro come Number TheoryBasic math Proofs
CONNECT through US
We usage cookies to give you the ideal experience on our website. Please click ok or Scroll down to use this site with cookies. Otherwise, check your internet browser settings to turn cookies turn off or discontinue using the site.OK!Cookie Policy