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Solving a logarithmic equation

The inverse of an exponential function is called a logarithmic function.

If y = log_b x,
then x = b^y where b > 0 and b 1

For example, 3 = log₂ 8
if, and only if 8 = 2³

Properties of logarithms

P1   log_b M X N = log_b M + log_b N

P2   log_b (M/N) = log_b M - log_b N

P3   log_b M^p = p X log_b M

P4   log_b 1 = 0

P5   log_b b = 1

P6   log_b b^x = x

N. B. By convention, log x designates log₁₀ x and
ln x designates the natural logarithm of x.

Example 1

Solve: x = log₃ 27

The first method to solve this equation is to express it in the form of an exponential equation.

Thus, x = log₃ 27

If, and only if 3^x = 27

and if 3^x = 3³

x = 3

Example 2

Solve; x = log₃ 27

Another method of solving this equation is to apply the various properties of logarithms, when possible.

Thus x = log₃ 27

x = log₃ 3³

x = 3 X log₃ 3 by P3

x = 3 X 1 by P5

x = 3

Example 3

Solve; x = log₂ (1/16)

Solution

Expressing this equation in an exponential form we see,

2^x = 1/16

2^x = 1/2⁴

2^x = 2^-4

x = -4

Example 4

Solve; 3^2x = 5 X 4^x

In this example, the method introduced in lesson 16 cannot be used since it is not possible to express the terms of the equation as powers of the same base.

Therefore, if we take the logarithm on each side of the equation, then;

3^2x = 5 X 4^x

log 3 ^2x = log 5 X 4^x

log 3 ^2x = log 5 + log 4^x by P1

2x X log 3 = log 5 + x X log 4 by P3

2x X log 3 - x X log 4 = log 5
x(2 X log 3 - log 4) = log 5

x =

= 1.985

Before proceeding to the questions, ensure that you have a pencil, some paper, and a calculator handy.

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