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# Exponents and Logarithmic Equations Help (page 2)

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#### Examples

Use Property 3 to rewrite the logarithms.

• log 4 3 x = x log 4 3

• log x 2 = 2 log x
• −3 log 8 = log 8 −3

### Three More Important Logarithm Properties Practice Problems - Set 1

#### Practice

Use Property 1 to rewrite the logarithms in Problems 1–6.

1. ln 59 t
2. log 0.10 y
3. log 30 148 x 2
4. log 6 3 + log 6 12
5. log 5 9 + log 5 10
6. log 5 + log 20

Use Property 2 to rewrite the logarithms in Problems 7–12.

1. log 7 2 − log 7 4
2. log 8 x − log 8 3

Use Property 3 to rewrite the logarithms in Problems 13–20.

1. In 5 x

1. log 5 6 t
2. 2 log 8 3
3. ( x + 6) log 4 3
4. log 16 10 2 x
5. −2 log 4 5

#### Solutions

1. ln 59 t = ln 59 + ln t
2. log 0.10 y = log 0.10 + log y = log 10 −1 + log y = −1 + log y
3. log 30 148 x 2 = log 30 148 + log 30 x 2
4. log 6 3 + log 6 12 = log 6 (3 · 12) = log 6 36 = log 6 6 2 = 2
5. log 5 9 + log 5 10 = log 5 (9 · 10) = log 5 90
6. log 5 + log 20 = log(5 · 20) = log 100 = log 10 2 = 2

1. log 5 6 t = − t log 5 6

2. 2 log 8 3 = log 8 3 2 = log 8 9

3. ( x + 6) log 4 3 = log 4 3 x +6

4. log 16 10 2 x = 2 x log 16 10

Sometimes we will need to use several logarithm properties to rewrite more complicated logarithms. The hardest part of this is to use the properties in the correct order. For example, which property should be used first on log ? Do we first use the third property or the second property? We will use the second property first. For the expression log , we would use the third property first.

Going in the other direction, we need to use all three properties in the expression log 2 9 − log 2 x + 3 log 2 y. We need to use the second property to combine the first two terms.

We cannot use the first property on until we have used the third property to move the 3.

#### Examples

Rewrite as a single logarithm.

• log 2 3 x − 4 log 2 y
• We need use the third property to move the 4, then we can use the second property.

• 3 log 4 x + 2 log 3 − 2 log y

• t ln 4 + ln 5
• t ln 4 + ln 5 = ln 4 t + ln 5 = ln(5 · 4 t )    (not ln 20 t )

Expand each logarithm.

### Three More Important Logarithm Properties Practice Problems - Set 2

#### Practice

For Problems 1−5, rewrite each as a single logarithm.

1. 1. 2 log x + 3log y
2. 2. log 6 2 x − 2 log 6 3
3. 3. 3 ln t − ln 4 + 2 ln 5
4. 4. t ln 6 + 2 ln 5

For Problems 6–10, expand each logarithm.

#### Solutions

1.  2 log x + 3 log y = log x 2 + log y 3 = log x 2 y 3

4. t ln 6 + 2 ln 5 = ln 6 t + ln 5 2 = ln[25(6 t )]

## More Logarithm Equations

With these logarithm properties we can solve more logarithm equations. We will use these properties to rewrite equations either in the form “log = log” or “log = number.” When the equation is in the form “log = log,” the logs cancel. When the equation is in the form “log = number,” we will rewrite the equation as an exponential equation. Instead of checking solutions in the original equation, we only need to make sure that the original logarithms are defined for the solutions.

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