1. Exponential and Logarithmic Equations

2. Using Natural Logarithms to Solve Exponential Equations
1. Isolate the exponential expression.
2. Take the natural logarithm on both sides of the equation.
3. Simplify using one of the following properties:
ln bx = x ln b or ln ex = x.
4. Solve for the variable.

3. Text Example Solve: 54x – 7 – 3 = 10
Solution We begin by adding 3 to both sides to isolate the exponential expression, 54x – 7. Then we take the natural logarithm on both sides of the equation.
54x – 7 – 3 = 10 This is the given equation.
54x – 7 = Add 3 to both sides.
ln 54x – 7 = ln 13 Take the natural logarithm on both sides.
(4x – 7) ln 5 = ln 13 Use the power rule to bring the exponent to the front.
4x ln 5 – 7 ln 5 = ln 13 Use the distributive property on the left side of the equation.
4x ln 5 = ln ln 5 Isolate the variable term by adding 7 ln 5 to both sides.
x = (ln 13)/(4 ln 5) + (7 ln 5)/(4 ln 5) Isolate x by dividing both sides by 4 ln 5.
The solution set is {(ln ln 5)/(4 ln 5)} approximately 2.15.

4. Text Example Solve: log4(x + 3) = 2.
Solution We first rewrite the equation as an equivalent equation in exponential form using the fact that logb x = c means bc = x.
log4 (x + 3) = 2 means 42 = x + 3
Now we solve the equivalent equation for x.
42 = x + 3 This is the equivalent equation.
16 = x + 3 Square 4.
13 = x Subtract 3 from both sides.
Check
log4 (x + 3) = 2 This is the logarithmic equation.
log4 (13 + 3) = 2 Substitute 13 for x.
log4 16 = 2
2 = 2
This true statement indicates that the solution set is {13}. ?

5. Example Solve 3x+2-7 = 27 Solution: 3x+2= 34 ln 3 x+2 = ln 34
(x+2) ln 3 = ln 34
x+2 = (ln 34)/(ln 3)
x+2 = 3.21
x = 1.21

6. Example Solve log 2 (3x-1) = 18 Solution: 2 18 = 3x-1 262,144 = 3x - 1

8. How long will it take 25,000 to grow to 500,000 at 9% annual interest compounded monthly?