1. Exponential & Logarithmic Equations
Exponential Equations with Like Bases
Exponential Equations with Different Bases
Logarithmic Equations

2. Exponential Equations with Like Bases
In an Exponential Equation, the variable is in the exponent. There may be one exponential term or more than one, like…
If you can isolate terms so that the equation can be written as two expressions with the same base, as in the equations above, then the solution is simple. or

3. Exponential Equations with Like Bases
Example #1 - One exponential expression.
1. Isolate the exponential expression and rewrite the constant in terms of the same base.
2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.

4. Exponential Equations with Like Bases
Example #2 - Two exponential expressions.
1. Isolate the exponential expressions on either side of the =. We then rewrite the 2nd expression in terms of the same base as the first.
2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.

5. Exponential Equations with Different Bases
The Exponential Equations below contain exponential expressions whose bases cannot be rewritten as the same rational number.
The solutions are irrational numbers, we will need to use a log function to evaluate them. or

6. Exponential Equations with Different Bases
Example #1 - One exponential expression.
1. Isolate the exponential expression.
2. Take the log (log or ln) of both sides of the equation.
3. Use the log rule that lets you rewrite the exponent as a multiplier.

7. Exponential Equations with Different Bases
Example #1 - One exponential expression.
4. Isolate the variable.

8. Exponential Equations with Different Bases
Example #2 - Two exponential expressions.
1. The exponential expressions are already isolated.
2. Take the log (log or ln) of both sides of the equation.
3. Use the log rule that lets you rewrite the exponent as a multiplier on each side..

9. Exponential Equations with Different Bases
Example #2 - Two exponential expressions.
4. To isolate the variable, we need to combine the ‘x’ terms, then factor out the ‘x’ and divide.

10. Logarithmic Equations
In a Logarithmic Equation, the variable can be inside the log function or inside the base of the log. There may be one log term or more than one. For example …

11. Logarithmic Equations
Example 1 - Variable inside the log function.
1. Isolate the log expression.
2. Rewrite the log equation as an exponential equation and solve for ‘x’.

12. Logarithmic Equations
Example 2 - Variable inside the log function, two log expressions.
1. To isolate the log expression, we 1st must use the log property to combine a difference of logs.
2. Rewrite the log equation as an exponential equation (here, the base is ‘e’).
3. To solve for ‘x’ we must distribute the ‘e’ and then collect the ‘x’ terms together and factor out the ‘x’ and divide.

13. Logarithmic Equations
Example 3 - Variable inside the base of the log.
1. Rewrite the log equation as an exponential equation.
2. Solve the exponential equation.