Exponents and Logarithmic Equations Help

Introduction to Exponents and Logarithmic Equations

For some logarithmic equations, a solution might be extraneous solution. That is, such a solution is a solution to the rewritten equations but not to the original equations. Some solutions to the rewritten equations will cause logarithms of 0 or of negative numbers. We can check them in the original equation to see which solutions are true solutions.

Examples

Solve for x .

log ₂ ( x ² + 3 x − 10) = 3
We will rewrite this as an exponent equation: 2 ³ = x ² + 3 x − 10 and solve for x .

x ² + 3 x − 10 = 8

x ² + 3 x − 18 = 0

( x + 6)( x − 3) = 0

The solutions are x = − 6 and x = 3. We will check them in the original equation.

Exponents and Logarithms Exponent and Logarithmic Equations (Part II)

The solutions to the original equation are x = −6 and x = 3.

log ₅ ( x ² + 5 x − 4) = log ₅ ( x + 1)
The logs cancel leaving x ² + 5 x − 4 = x + 1.

x ² + 5 x − 4 = x + 1

x ² + 4 x − 5 = 0

( x + 5)( x − 1) = 0

The solutions are x = − 5 and x = 1. We cannot allow x = − 5 as a solution because log ₅ (−5 + 1) is not defined. We need to check x = 1.

log ₅ (1 ² + 5(1) − 4) = log ₅ (1 + 1) is true

The solution is x = 1.

Exponents and Logarithmic Equations Practice Problems

Practice

Solve for x .

ln( x ² + x − 20) = ln(3 x + 4)
log ₄ (2 x ² − 3 x + 59) = 3

Solutions

ln( x ² + x − 20) = ln(3 x + 4)

x ² + x − 20 = 3 x + 4

x ² − 2 x − 24 = 0

( x − 6)( x + 4) = 0

The solutions are x = 6 and x = −4. Because ln[3(−4) + 4] is not defined, we only need to check x = 6.

ln(6 ² + 6 − 20) = ln[3(6) + 4] is true.

The only solution is x = 6.
log ₄ (2 x ² − 3 x + 59) = 3

Three More Important Logarithm Properties

The following three logarithm properties come directly from the exponent properties Exponents and Logarithms Three More Important Logarithm Properties .

log _b mn = log _b m + log _b n
log _b m ^t = t log _b m

We will see why Property 1 works. Let x = log _b m and y = log _b n . Rewriting these equations as exponential equations, we get b ^x = m and b ^y = n . Multiplying m and n , we have mn = b ^x · b ^y = b ^{x + y} . Rewriting the equation mn = b ^{x + y} as a logarithmic equation, we get log _b mn = x + y . Because x = log _b m and y = log _b n , log _b mn = x + y becomes log _b mn = log _b m + log _b n .

Examples

Use Property 1 to rewrite the logarithms.

log ₄ 7 x = log ₄ 7 + log ₄ x
ln 15 t = ln 15 + ln t
log ₆ 19 t ² = log ₆ 19 + log ₆ t ²
log 100 y ⁴ = log 10 ² + log y ⁴ = 2 + log y ⁴

log ₉ 3 + log ₉ 27 = log ₉ 3(27) = log ₉ 81 = 2

Use Property 2 to rewrite the logarithms.

Exponents and Logarithms Three More Important Logarithm Properties

The exponent property Exponents and Logarithms Three More Important Logarithm Properties allows us to apply the third logarithm property to roots as well as to powers. The third logarithm property is especially useful in science and business applications.

Examples

Use Property 3 to rewrite the logarithms.

log ₄ 3 ^x = x log ₄ 3

Exponents and Logarithms Three More Important Logarithm Properties

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.

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

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

Exponents and Logarithms Three More Important Logarithm Properties

log ₇ 2 − log ₇ 4
log ₈ x − log ₈ 3

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

In 5 ^x

Exponents and Logarithms Three More Important Logarithm Properties

log ₅ 6 ^{− t}
2 log ₈ 3
( x + 6) log ₄ 3
log ₁₆ 10 ^{2 x}
−2 log ₄ 5

Solutions

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

Exponents and Logarithms Three More Important Logarithm Properties

log ₅ 6 ^{− t} = − t log ₅ 6
2 log ₈ 3 = log ₈ 3 ² = log ₈ 9
( x + 6) log ₄ 3 = log ₄ 3 ^{x +6}
log ₁₆ 10 ^{2 x} = 2 x log ₁₆ 10

Exponents and Logarithms Three More Important Logarithm Properties

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 Exponents and Logarithms Three More Important Logarithm Properties ? 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.

Exponents and Logarithms Three More Important Logarithm Properties

We cannot use the first property on Exponents and Logarithms Three More Important Logarithm Properties until we have used the third property to move the 3.

Exponents and Logarithms Three More Important Logarithm Properties

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.

Exponents and Logarithms Three More Important Logarithm Properties

3 log 4 x + 2 log 3 − 2 log y

Exponents and Logarithms Three More Important Logarithm Properties

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.

Exponents and Logarithms Three More Important Logarithm Properties

Three More Important Logarithm Properties Practice Problems - Set 2

Practice

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

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

Exponents and Logarithms Three More Important Logarithm Properties

For Problems 6–10, expand each logarithm.

Exponents and Logarithms Three More Important Logarithm Properties

Solutions

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

Exponents and Logarithms Three More Important Logarithm Properties

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

Exponents and Logarithms Three More Important Logarithm Properties

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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Exponents and Logarithmic Equations Help (page 2)

Introduction to Exponents and Logarithmic Equations

Examples

Exponents and Logarithmic Equations Practice Problems

Practice

Solutions

Three More Important Logarithm Properties

Examples

Examples

Three More Important Logarithm Properties Practice Problems - Set 1

Practice

Solutions

Examples

Three More Important Logarithm Properties Practice Problems - Set 2

Practice

Solutions

More Logarithm Equations

Examples

More Logarithm Equations Practice Problems

Practice

Solutions

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