Logarithms

Logarithms

Logarithmic and Exponential Equations

Learning objective

• use logarithms to solve equations in

  which the unknown appears in indices

Solving logarithmic equations

• Since log_ax only exists for x > 0 and for a > 0 and a≠1, it is essential when solving equations involving logs that all roots are checked in the original equation

Example 1

• 2 log₈(x+2)=log₈(2x+19)

• log₈(x+2)²=log₈(2x+19)

• (x+2)² = 2x+19

  x²+4x+4=2x+19

• x²+2x-15=0

• (x-3)(x+5)=0

• x=3 or x=-5

• So, x=3

Example 2

• 4  log_x2−log_x4=2

• log_x2⁴−log_x2²=2

• log_x2⁴⁻²=2

• log_x2²=2

• log_x4=2

• x²=4 -------> x=2 or x=-2

• So, x = 2

Solving exponential equations

• Solve, giving your answers correct to 3 significant figures.

  5^2x+1=7

• log  5^2x+1=log  7

• (2x+1)log  5=log  7

• 2x  log  5+log  5=log  7

• x=log  7−log  5/2  log  5

• x=0.105

Example 3

• Solve, giving your answers correct to 3 significant figures.

  3^2x = 4^x+5

• log  3^2x = log  4^x+5

• 2x  log  3 = (x+5)log  4

• 2x  log  3 = x  log  4+5  log  4

• x(2  log  3−log  4)=5  log  4

• x=5  log  4/2  log  3−log  4

• x=8.55