**Topic 5-4: Solving Logarithmic Equations **

A logarithmic equation is an equation involving one

or more logarithmic expressions.

Logarithmic equations can be divided into two

families :

1. Those equations that can be simplified into the form

of a logarithmic expression equal to a number .

2. Those equations that can be simplified into the form

of two logarithmic equations (same base) equal to

each other.

To solve the first case, apply the following steps :

1. Use the laws of logarithms , as necessary, to

combine logarithms .

2. Rewrite the logarithm equation into exponential

form.

3. Solve using any appropriate algebraic and/or

arithmetic methods.

4. Check your solutions. Like radical equations ,

logarithmic equations can (and often do) produce

extraneous solutions.

Ex. 1 Solve.

Ex. 2 Solve.

Ex. 3 Solve.

Ex. 4 Solve.

If the equation is simplified (using the laws of

logarithms) in such a way both sides have a

logarithmic expression, consider the limited

possibilities for making the equation true...

Ex. 1 Solve.

Ex. 2 Solve.

log(x + 4) + log(x + 5) −log(1− x) = log(x + 7)

**Topic 5-5: Solving Exponential Equations**

An exponential equation is an equation involving

one or more exponential expressions.

Basic process to solve most exponential

equations:

1. Isolate an exponential expression on one side .

2. Take the natural logarithm (or common log) of both

sides.

3. Use the laws of logarithms to rewrite the exponential

expression so that no variable remains in the

exponent.

4. Apply basic algebraic and arithmetic manipulation to

solve for x.

5. Use the laws of logarithms to simplify the solution

and approximate the solution.

6. Check your solution.

With regards to step 5, I will expect you to further rewrite

the solution into a single logarithmic expression and

approximate five places after the decimal .

Ex. 1 Solve.

6^{3x} = 100

Ex. 2 Solve.

4 + 3^{x-2}= 16

Ex. 3 Solve.

5^{2x-1} = 18

Ex. 4 Solve.

10^{2x+1} = 500

Ex. 5 Solve.

Exponential equations involving 2 exponential

expressions

If the bases are equal or the bases are both powers

of the same constant, solving the equation is fairly

easy.

Ex. 1 Solve.

Ex. 2 Solve.

If the bases are neither equal nor related, revert

back to the previously covered process for solving

exponential equations but more or less ignore step

1.

Ex. 1 Solve.

Ex. 2 Solve.

Ex. 3 Solve.

Exponential equations solved by substitution and

factoring

The important skill to remember here is

that

Ex. 1 Solve.

Ex. 2 Solve.