To solve equations with exponents and logarithms apply all the rules and properties of exponents and logs.

 

Example

 

Solve for x in e^x = 15

 

The first thing to do is to take the natural log of both sides. This will eventually get rid of the exponent.

 

ln e^x = ln 15

 

ln e^x = x by the properties of natural logs.

 

x = ln 15

 

Find ln 15 with a calculator.

 

x = 2.70805

 

Answer!

 

 

 

 

 

 

Example

 

Solve for x in 4^x = 115

 

Take the log of both sides.

 

log 4^x = log 115

 

Use properties of logs to pull the exponent out front.

 

x log 4 = log 115

 

Divide both sides by log 4.

 

x =

 

Use a calculator to find log 115 and log 4.

x =

 

Divide.

 

x = 3.422745

 

Answer!

 

Note- In the previous example the base was not shown on .It is understood that when there is not a base stated, the base is 10.

 

 

 

 

 

Example

 

Solve for x in 2^4x = 35

 

Take the log of both sides.

 

log 2^4x = log 35

 

Use the property of logs to pull the 4x out front.

 

4x log 2 = log 35

 

Divide both sides by log 2.

 

4x =

 

 

Calculate .

 

 

4x = 5.12928

 

Divide by 4.

 

x = 1.28232

 

Answer!

 

 

 

 

Example

Solve for x in log₃2x = 4

 

Put in exponential form.

3⁴ = 2x

 

Find 3⁴.

81 = 2x

 

Divide by 2.

40.5 = x

 

Turn the equation around.

x = 40.5

 

Answer!

 

 

 

 

 

Example

 

Solve for x in 5^x+1 = 3^2x

 

Take the log, or natural log, of both sides.

 

ln 5^x+1 = ln 3^2x

 

Apply the exponents rule to both sides.

 

(x +1) ln 5 = 2x ln 3

 

Find the ln 5 and ln 3.

 

(x + 1)(1.60944) = 2x(1.09861)

 

Multiply to get rid of parentheses.

1.60944x + 1.60944 = 2.19722x

 

Solve for x by first subtracting 2.19722x from both sides.

 

-.58778x + 1.60944 = 0

 

Subtract 1.60944 from both sides.

 

-.58778x = -1.60944

 

Divide both sides by -.58778.

x = 2.73817

 

Answer!

 

 

 

 

 

 

Example

 

ln x + ln 2x = 4

 

Apply the rule of logarithmic multiplication only in reverse order.

 

ln x(2x) = 4

 

Now this step is new and kind of tricky. Use both sides as exponents of e. The reason will be explained in the next step.

 

e^{ln x(2x)} = e⁴

 

The reason to did this is because e^{ln y} = y. Apply that idea here.

 

x(2x) = e⁴

 

Find e⁴, and simplify the left side by multiplying.

 

2x² = 54.59815

 

Divide both sides by 2.

 

x² = 27.29908

 

Take the square root of both sides.

x = 5.22485

 

Answer! Only use the positive answer because the ln of a negative number does not exist.