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The Change of Base Formula Help

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By — McGraw-Hill Professional
Updated on Oct 4, 2011

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.

 

Exponents and Logarithms The Change of Base Formula

 

This means that 2 2.321928095 is very close to 5.

We just proved that Exponents and Logarithms The Change of Base Formula . Replace 2 with b , 5 with x , and 10 with a and we have the change of base formula.

Change of Base Formula

Exponents and Logarithms The 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.

Exponents and Logarithms The Change of Base Formula

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 Exponents and Logarithms The Change of Base Formula , the equation can be written as Exponents and Logarithms The Change of Base Formula.

Exponents and Logarithms The Change of Base Formula

Example

  • Exponents and Logarithms The Change of Base Formula
  • Rewriting this as a logarithm equation, we get Exponents and Logarithms The Change of Base Formula . Now we can use the change of base formula.

Exponents and Logarithms The Change of Base Formula

The Change of Base Formula Practice Problems

Practice

Evaluate the logarithms. Give your solution accurate to four decimal places.

  1. log 6 25
  2. log 20 5

Solve for x . Give your solutions accurate to four decimal places.

  1. 3 x +2 = 12
  2. 15 3 x −2 = 10
  3. 24 3 x +5 = 9

Solutions

Exponents and Logarithms The Change of Base Formula

3. Rewrite 3 x +2 = 12 as a logarithm equation: x + 2 = log 3 12

   Exponents and Logarithms The Change of Base Formula

4. Rewrite 15 3 x −2 = 10 as a logarithm equation: 3 x − 2 = log 15 10

   Exponents and Logarithms The Change of Base Formula

5. Rewrite 24 3 x +5 = 9 as a logarithm equation: 3 x + 5 = log 24 9.

   Exponents and Logarithms The Change of Base Formula

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 .

  1. Distribute the logarithms.
  2. Collect the x terms on one side of the equation and the non- x terms on the other side.
  3. Factor x .
  4. 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.

Exponents and Logarithms The Change of Base Formula

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.

Exponents and Logarithms The Change of Base Formula

 

  • 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.

     Exponents and Logarithms The Change of Base Formula

Exponential Equations with Exponents on Both Sides Practice Problems

Practice

Solve for x . Give your solutions accurate to four decimal places.

  1. 4 x = 5 x −1
  2. 6 2 x = 8 3 x −1
  3. 10 2− x = 5 x +3

Solutions

1. Take the natural log of each side of 4 x = 5 x −1 .

  Exponents and Logarithms The Change of Base Formula

2. Take the natural log of each side of 6 2 x = 8 3 x −1 .

  Exponents and Logarithms The Change of Base Formula

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.

  Exponents and Logarithms The Change of Base Formula

Find practice problems and solutions for these concepts at: Exponents and Logarithms Practice Test.

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