Exponential and logarithmic equations
Solving exponential and logarithmic equations
Exponential 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  3x - 1 = 81.
Solution:             3x - 1 = 81
3x - 1 = 34
x - 1 = 4          =>     x = 5.
Example:   Solve     0.25x = 43x - 2.
 Solution:
Example:   Solve     43x + 2 = 64 · 22x - 4.
Solution:               42 · 43x = 43 · 22(x - 2) | ¸ 42
43x = 41 + x - 2
3x = x - 1
2x = - 1             =>     x = - 1/2.
Example:   Solve     4x + 2 + 4x + 1 + 4x -1  = 3x + 3 + 3x + 2.
 Solution:
Example:       Solve     4x - 2 = 5x.
 Solution:
 Example:   Solve
 Solution:
Example:   Solve    3 · 4x + 2 · 9x  = 5 · 6x.
 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 = 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:
Pre-calculus contents F