Exponential and logarithmic functions and equations

Exponential and Logarithmic Functions and Equations

Exponential and logarithmic equations

Solving exponential and logarithmic equations

Exponential equations

Logarithmic equations

Exponential and logarithmic equations

Exponential equations

In exponential equations the variable that has to be solved for is in the exponent.

To solve an exponential equation, rewrite the given equation to get all powers (exponentials) with the same base, or use logarithms when solving the exponential equation.

Example: Solve 3^{x
- 1} = 81.

Solution: 3^{x
- 1} = 81

3^{x
- 1} = 3⁴

x - 1 = 4 => x = 5.

Example: Solve 0.25^x = 4^3x^-².

Solution:

Example: Solve 4^3x⁺² = 64 · 2²^x^-⁴.

Solution: 4² · 4^3x = 4³ · 2²⁽^x^-²⁾| ¸ 4²

4^3x = 4^{1 +}^x^-²

3x = x - 1

2x = - 1 => x = - 1/2.

Example: Solve 4^x⁺² + 4^x⁺¹+ 4^x^-¹ = 3^x⁺³ + 3^x⁺².

Solution:

Example: Solve 4^x^-² = 5^x.

Solution:

Example: Solve

Solution:

Example: Solve 3 · 4^x + 2 · 9^x= 5 · 6^x.

Solution:

Logarithmic equations

As logarithmic equations contain a logarithm of variable quantity, we use rules and properties of logarithms to solve a logarithm equation.

Example: Solve log x = -2.

Solution: log x = log10^-2 => x = 10^-2 = 0.01.

Example: Solve log₂ (log₃ x) = 1.

Solution: log₃ x = 2¹ => x = 3² = 9.

Example: Solve log (x + 5) - log (2x - 3) = 2 · log 2.

Solution:

Example: Solve log (log x) + log (log x³ - 2) = 0.

Solution:

Example: Solve x ^{log x}^{-
2} = 1000.

Solution:

College algebra contents D