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# Logarithms Help

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## Introduction to Logarithms

A common question for investors is, “How long will it take for my investment to double?” If \$1000 is invested so that it earns 8% interest, compounded annually, how long will it take to grow to \$2000? To answer the question using the compound growth formula, we need to solve for t in the equation 2000 = 1000(1.08) t . We will divide both sides of the equation by 1000 to get 2 = (1.08) t . Now what? It does not make sense to “take the t th root” of both sides. We need to use logarithms. In mathematical terms, the logarithm and exponent functions are inverses. Logarithms (or logs ) are very useful in solving many science and business problems.

### Converting Logarithmic Equations to Exponential Equations

The logarithmic equation log a x = y is another way of writing the exponential equation a y = x . Verbally, we say, “log base a of x is (or equals) y .” For “log a x , we say,” (the) log base a of x .

#### Examples

Rewrite the logarithmic equation as an exponential equation.

• log 3 9 = 2
• The base of the logarithm is the base of the exponent, so 3 will be raised to a power. The number that is equal to the log is the power, so the power on 3 is 2.

log 3 9 = 2 rewritten as an exponent is 3 2 = 9

• The base is 2 and the power is −3.

• The base is 9 and the power is .

9 = 3

### Converting Exponential Equations to Logarithmic Equations

Now we will work in the other direction, rewriting exponential equations as logarithmic equations. The equation 4 3 = 64 written as a logarithmic equation is log 4 64 = 3.

#### Examples

• 3 4 = 81

The base of the logarithm is 3, and we are taking the log of 81. The equation rewritten as a logarithmic equation is log 3 81 = 4

• a 3 = 4

The base is a , and we are taking the log of 4. The equation rewritten as a logarithmic equation is log a 4 = 3.

The base is 8, and we are taking the log of 4. The equation rewritten as a logarithmic equation is log 8

## Rewriting Logarithmic Equations Practice Problems

#### Practice

For Problems 1-5, rewrite the logarithmic equations as exponential equations. For Problems 6-12 rewrite the exponential equations as logarithmic equations.

1. log 4 16 = 2
2.
3. log e 2 = 0.6931
4. log ( x +1) 9 = 2
5.
6. 5 2 = 25
7. 4 0 = 1
8.
9. 125 1/3 = 5
10. 10 −4 = 0.0001
11. e 1/2 = 1.6487
12. 8 x = 5

#### Solutions

1. log 4 16 = 2 rewritten as an exponential equation is 4 2 = 16
2.   rewritten as an exponential equation is
3. log e 2 = 0.6931 rewritten as an exponential equation is e 0.6931 = 2
4. log ( x +1) 9 = 2 rewritten as an exponential equation is ( x + 1) 2 = 9
5.   rewritten as an exponential equation is
6. 5 2 = 25 rewritten as a logarithmic equation is log 5 25 = 2
7. 4 0 = 1 rewritten as a logarithmic equation is log 4 1 = 0
8.   rewritten as a logarithmic equation is
9. 125 1/3 = 5 rewritten as a logarithmic equation is
10. 10 −4 = 0.0001 rewritten as a logarithmic equation is log 10 0.0001 = − 4
11. e 1/2 = 1.6487 rewritten as a logarithmic equation is
12. 8 x = 5 rewritten as a logarithmic equation is log 8 5 = x

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