Logarithmic Equations

These are my notes on logarithmic equations in Algebra.

Exponential equations are used to model rapid growth but logarithmic equations are used for modeling slower growth. 

An equation where one or more variables occur in the exponent of an expression is called an exponential equation. Exponential equations occur in a variety of applications and can be solved symbolically, graphically, and numerically.

Example 1

Solve \( 10^{x+2} = 10^{3x} \)

Take the common log on each side of the equation

\( \log 10^{x+2} = \log 10^{3x} \)

Apply the inverse property of logarithms

\( x+2 = 3x \)

Subtract an x from each side to isolate the larger x value on the right

\( 2 = 2x \)

Divide each side of the equation by 2

\( \boxed{x = 1} \)

Example 2

Solve \( 5(1.2)^{x} + 1 = 26 \)

Subtract 1 from each side of the equation

\( 5(1.2)^{x} = 25 \)

Divide each side of the equation by 5

\( (1.2)^{x} = 5 \)

Take the common log of each side

\( \log (1.2)^{x} = \log 5 \)

Apply the exponent property

\( x \log (1.2) = \log 5 \)

Divide each side by \( \log (1.2) \)

\( x = \frac{\log 5}{\log 1.2} \)

\(\boxed{ x = 8.8} \)

Example 3

Solve \( \frac{1}{4}^{x-1} = \frac{1}{10} \)

Take the common logarithm of each side

\( \log \frac{1}{4}^{x-1} = \log \frac{1}{10} \)

Apply the exponent property

\( (x-1) \log \frac{1}{4} = \log \frac{1}{10} \)

Divide each side by \( \log \frac{1}{4} \)

\( (x-1) = \frac{\log \frac{1}{10}}{\log \frac{1}{4}} \)

Add 1 to each side

\( x = 1 + \frac{\log \frac{1}{10}}{\log \frac{1}{4}} \)

\( \boxed{x = 2.6} \)

Example 4

Solve \( 5^{x-3} = e^{2x} \)

Take the natural logarithm of each side

\( \ln 5^{x-3} = \ln e^{2x} \)

Use the inverse property

\( (x-3) \ln 5 = 2x \)

Use the distributive property

\( x \ln 5 - 3 \ln 5 = 2x \)

Subtract \( 2x \) from both sides

Add \( 3 \ln 5 \) to both sides

\( x \ln 5 - 2x = 3 \ln 5 \)

Factor out the x

\( x(\ln 5 - 2) = 3 \ln 5 \)

Divide both sides by \( \ln 5 - 2 \)

\( x = \frac{3 \ln 5}{\ln 5 - 2} \)

\(\boxed{x = -12.36} \) 

Logarithmic equations contain logarithms. Like exponential equations, logarithmic equations also occur in applications. To solve a logarithmic equation, we use the inverse property of logarithms.

Example 5

Solve \( 3 \log_3 x = 12 \)

Divide each side by 3

\( \log_3 x = 4 \)

Exponentiate each side with base 3

\( 3^{\log_3 x} = 3^{4} \)

Apply the inverse property

\( \boxed{x = 81} \)

Like exponential equations, logarithmic equations can occur in many forms.

Example 6

Solve \( \log (2x + 1) = 2 \)

Exponentiate each side of the equation with base 10

\( 10^{\log (2x + 1)} = 10^{2} \)

Apply inverse property

\( 2x + 1 = 100 \)

Subtract 1 from each side

\( 2x = 99 \)

Divide each side by 2

\( \boxed{x = 49.5} \)

Example 7

Solve \( \log_2 4x = 2 - \log_2 x \)

Add \( \log_2 x \) to both sides

\( \log_2 4x + \log_2 x = 2 \)

Apply the additive property

\( \log_2 4x^{2} = 2 \)

Exponentiate each side of the equation by base 2

\( 2^{\log_2 4x^{2}} = 2^{2} \)

Apply the inverse property

\( 4x^{2} = 4 \)

Divide each side by 4

\( x^{2} = 1 \)

\( \boxed{x = 1 \text{ or} -1} \)

Example 8

Solve \( 2 \ln (x+1) = \ln (1-2x) \)

Apply the exponent property

\( \ln (x+1)^{2} = \ln (1-2x) \)

Exponentiate with base e

\( e^{\ln(x+1)^{2}} = e^{\ln(1-2x)} \)

Apply inverse property

\( (x+1)^{2} = (1-2x) \)

Expand the binomial

\( x^{2}+2x+1 = 1-2x \)

Combine terms

\(x^{2}+4x = 0 \)

Factor out an x

\( x(x+4) = 0 \)

\(\boxed{ x=0 \text{ or } x=-4} \)

At some point in the process of solving an exponential equation, we often take a logarithm of each side of the equation. Similarly, when solving a logarithmic equation, we often exponentiate each side of the equation.