You get a message dropped through a crack in your window in the middle of the night. It reads, "I am ready to lend a hand in the fight against the evil Expo. Meet me in the park at noon tomorrow." When you get there the next day, it's none other than Ms. Log Arithm! She greets you warmly and jumps right to business.

Practice:

Example 1
Solve the exponential equation 3^{x} – 5 = 15y + 5 for x

First, isolate the term with x in it: 3^{x} = 15y + 10

Then take the base-3 logarithm of both sides:

log_{3} (3^{x} ) = log_{3} (15y + 10)

Simplify, using the rules of logarithms: x = log_{3} (15y + 10)

x = log_{3} (15y +10)

Show Next Step

Example 2
Change the base of log_{4} 20 to base 10 and solve to 4 decimal places

The change of base formula is log_{a} x = log_{b} x / log_{b} a :

Show Next Step

Example 3
Find the inverse of the function: y = (7^{x} + 5) ^{1/5}

Take the base-y log of both sides:

log_{y} y = log_{y} (7^{x} + 5) ^{1/5}

The left side cancels:

1 = log_{y} (7^{x} + 5) ^{1/5}

Pull out the exponent using the exponent in log rule:

Multiply both sides by 5: 5 = log_{y} (7^{x} + 5)

Exponentiate both sides with a base of y : y ^{5} = 7^{x} + 5

Isolate the term with x : y ^{5} – 5 = 7^{x}

Take the base-7 log:

log_{7} (y ^{5} – 5) = log_{7} 7^{x}

The right side cancels: log_{7} (y^{5} – 5) = x

Switch x and y : y = log_{7} (x ^{5} – 5)

y = log_{7} (x ^{5} – 5)

Show Next Step

Example 4
Show to 3 decimal places that log_{4} 64 = log_{7} 343 using base-10 logarithms

First, use the change of base formula on both sides:

Solve each individually:

Simplify: 3 = 3

True!

Show Next Step

Example 1
Solve the following exponential equation for y : 7^{y + 1} = 3x + 3

Gimme a Hint
Hint

Take the log of both sides, but which log?

Show Answer Answer

y = log_{7} (3x + 3) – 1

Example 2
Find x in the following exponential equation. Do not simplify to decimals: 4 × (e ^{x} ) + 10 = 20

Gimme a Hint
Hint

Isolate x in exponent first.

Show Answer

Example 3
What value of x fulfills the following equation: log_{10} 10 = x

Gimme a Hint
Hint

What does x represent in the exponential form?

Show Answer

Example 4
What value of x fulfills the following equation: log_{3} 1 = x

Gimme a Hint
Hint

Any (arguably non-zero) number can be raised to a particular value to yield one. What power is it?

Show Answer

Example 5
Is the following logarithmic equation valid for any one value of x : log_{10} x = -3

Gimme a Hint
Hint

Can a base have a negative exponent?

Show Answer Answer

Yes. x is simply

Example 6
Simplify the following exponential equation for y : e ^{y + 4} – 15 = x^{2}

Gimme a Hint
Hint

Isolate y in exponent first.

Show Answer Answer

y = ln(x ^{2} + 15) – 4

Example 7
Solve the logarithmic equation for . Do not simplify to decimals.

Answer

or

Example 8
Evaluate the following log equation for x = 3, to three decimal places: y = log_{3} (x ^{2} ) + log_{3} (2x + 3) + log_{3} 8x

Gimme a Hint
Hint

Try separating or simplifying logs before solving them.

Show Answer

Example 9
Evaluate the following log equation for x = 5, to three decimal places: y = ln_{3x} + ln_{ex}

Gimme a Hint
Hint

One ln can be simplified first.

Show Answer

Example 10
Evaluate the following log equation to three decimal places using only base-10 logarithmic functions: y = log_{3} 13

Gimme a Hint
Hint

Use the change of base formula.

Show Answer Answer

Example 11
Write the following logarithmic equation in base 10 form: log_{4} 17

Gimme a Hint
Hint

Use the change of base formula.

Show Answer Answer

Example 12
Write the following logarithmic equation in base 10 form: ln 4x

Gimme a Hint
Hint

Use the change of base formula.

Show Answer Answer