Introduction to The Change of Base Formula
There are countless bases for logarithms but calculators usually have only two logarithms—log and ln. How can we use our calculators to approximate log _{2} 5? We can use the change of base formula but first, let us use logarithm properties to find this number. Let x = log _{2} 5. Then 2 ^{x} = 5. Take the common log of each side.
This means that 2 ^{2.321928095} is very close to 5.
We just proved that . Replace 2 with b , 5 with x , and 10 with a and we have the change of base formula.
Change of Base Formula
This formula converts a logarithm with old base b to new base a . Usually, the new base is either e or 10.
Example
 Evaluate log _{7} 15. Give your solution accurate to four decimal places.
Solving Equations using the Change of Base Formula
The change of base formula can be used to solve equations like 4 ^{2 x +1} = 8 by rewriting the equation in logarithmic form and using the change of base formula. The equation becomes log _{4} 8 = 2 x + 1. Because , the equation can be written as .
Example

Rewriting this as a logarithm equation, we get . Now we can use the change of base formula.
The Change of Base Formula Practice Problems
Practice
Evaluate the logarithms. Give your solution accurate to four decimal places.
 log _{6} 25
 log _{20} 5
Solve for x . Give your solutions accurate to four decimal places.
 3 ^{x +2} = 12
 15 ^{3 x −2} = 10
 24 ^{3 x +5} = 9
Solutions
3. Rewrite 3 ^{x +2} = 12 as a logarithm equation: x + 2 = log _{3} 12
4. Rewrite 15 ^{3 x −2} = 10 as a logarithm equation: 3 x − 2 = log _{15} 10
5. Rewrite 24 ^{3 x +5} = 9 as a logarithm equation: 3 x + 5 = log _{24} 9.
Solving Exponential Equations with Exponents on Both Sides
When both sides of an exponential equation have an exponent, we will use another method to solve for x . We will take either the natural log or the common log of each side and will use the third logarithm property to move the exponents in front of the logarithm. Once we have used the third logarithm property, we will perform the following steps to find x .
 Distribute the logarithms.
 Collect the x terms on one side of the equation and the non x terms on the other side.
 Factor x .
 Divide both sides of the equation by x ’s coefficient (found in Step 3).
Examples
 3 ^{2 x} = 2 ^{x +1}

We will begin by taking the natural log of each side.
Now we want both terms with an x in them on one side of the equation and the term without x in it on the other side. This means that we will move x ln 2 to the left side of the equation.
 10 ^{x +4} = 6 ^{3 x −1}

Because one of the bases is 10, we will use common logarithms. This will simplify some of the steps. We will begin by taking the common log of both sides.
Exponential Equations with Exponents on Both Sides Practice Problems
Practice
Solve for x . Give your solutions accurate to four decimal places.
 4 ^{x} = 5 ^{x −1}
 6 ^{2 x} = 8 ^{3 x −1}
 10 ^{2− x} = 5 ^{x +3}
Solutions
1. Take the natural log of each side of 4 ^{x} = 5 ^{x −1} .
2. Take the natural log of each side of 6 ^{2 x} = 8 ^{3 x −1} .
3. Take the common log of each side of 10 ^{2− x} = 5 ^{x +3} . This lets us use the fact that log 10 ^{2− x} = 2− x.
Find practice problems and solutions for these concepts at: Exponents and Logarithms Practice Test.
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