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:

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

First, isolate the term with *x* in it: 3^{x} = 15*y* + 10 Then take the base-3 logarithm of both sides: log_{3}(3^{x}) = log_{3}(15*y* + 10) Simplify, using the rules of logarithms: *x* = log_{3}(15*y* + 10) *x* = log_{3}(15*y*+10)
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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*:
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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)
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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! | |

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

Hint

Take the log of both sides, but which log?

Answer

*y* = log_{7} (3*x* + 3) – 1

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

Hint

Isolate *x* in exponent first.

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

Hint

What does *x* represent in the exponential form?

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

Hint

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

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

Hint

Can a base have a negative exponent?

Answer

Yes. *x* is simply

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

Hint

Isolate y in exponent first.

Answer

*y* = ln(*x*^{2} + 15) – 4

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

Answer

or

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

Hint

Try separating or simplifying logs before solving them.

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

Hint

One ln can be simplified first.

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

Hint

Use the change of base formula.

Answer

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

Hint

Use the change of base formula.

Answer

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

Hint

Use the change of base formula.

Answer