1. Solve for x: 1) xln2 = ln3 2) (x – 1)ln4 = 2
Warm-up:
Solve for x:
1) xln2 = ln3 2) (x – 1)ln4 = 2
HW Page 335 EVEN (2-6, 12 – 36, 52, 56 – 74)

2. Exponential & Logarithmic Equations
Objective:
Exponential Equations with Like Bases
Exponential Equations with Different Bases
Logarithmic Equations with variable inside the log function
Logarithmic Equations with variable inside the base of the log

3. 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…
The goal is to isolate terms so that the equation can be written as two expressions with the same base. or

4. Exponential Equations with Like Bases
Example #1 - One exponential expression.
1. Isolate the exponential expression.
2. Rewrite 9 to set each side to the same base.
3. Exponents must be equal

5. Exponential Equations with Like Bases
Example #2 - Two exponential expressions.
1. Rewrite the 2nd expression to set it to the same base as the first.
2. Exponents must be equal

6. 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

7. Exponential Equations with Different Bases
Example #1 - One exponential expression.
1. Isolate the exponential with the variable.
2. Take the log (log or ln) of both sides of the equation.
3. Inverse property of logs.
4. Calculator.

8. Exponential Equations with Different Bases
Example #2 - 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 power rule
4. Isolate the variable.

9. Exponential Equations with Different Bases
Example #3 - Two exponential expressions.
1. Take the log (log or ln) of both sides of the equation.
2. Use the log power rule
3. To isolate the variable, we need to combine the ‘x’ terms, then factor out the ‘x’ and divide.

10. Exponential Equations with Different Bases
Example #4 - Two exponential expressions.
1. Put in quadratic form
2. Factor
3. Zero Product Property

11. 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 …

12. Logarithmic Equations
Example 1 - Variable inside the log function.
Rewrite the log equation as an exponential equation with 4 as a base.

13. Logarithmic Equations
Example 2 - Variable inside the log function.
1. Isolate the natural log with variable
2. Rewrite the natural log equation as an exponential equation with e as a base.

14. Logarithmic Equations
Example 3 - Variable inside the log function.
1. Isolate the log expression.
2. Rewrite the log equation as an exponential equation and solve for ‘x’.

15. Logarithmic Equations
Example 4 - 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.

16. Logarithmic Equations
Example 5 - Variable inside the base of the log.
1. Rewrite the log equation as an exponential equation.
2. Solve the exponential equation.

17. log10(x+4) – log10x = log10(x + 2)
Sneedlegrit:
Solve for x.
log10(x+4) – log10x = log10(x + 2)
HW Page 335 EVEN (2-6, 12 – 36, 52, 56 – 74)