Exponential/Logarithmic Equations

Rewriting Base

100

Solve: 5^x = 125

x = 3

200

Solve: 3^x = 81^(x+3)

x = -4

300

Solve: 16^x = 8^(x+4)

x = 12

400

What do we need to do when we cannot rewrite the base?

Log both sides

500

Solve: 4^(x+1) = 8^(2x-1)

x = 5/4

Log on One Side

100

Solve: log₄x = 2

x = 16

200

Solve: log(x-2) = 1

x = 12

300

Solve: log₂(x+1) = 5

x = 31

400

Solve: log₃(x^2 - 4) = 2

x = √13

500

Why is the purpose of rewriting our logarithmic equations?

To get them into an exponential equation and solve for x.

Log Both Sides

100

Solve: 3^x = 20

x = 2.73

200

Solve: 9^x = 14

x = 1.2

300

Solve: 5^(x+1) = 30

x = 1.46

400

Solve: 2^(x+2) = 7^x

x = 1.1

500

Why does applying the log function to both sides of our equation work?

Logs are the inverses of exponential functions, and it allows us to bring down our exponent

Properties of Logs

100

How can I condense log₂(4) + log₂(5x)?

log₂(20x)

200

How can I condense log₃4 - log₃(x-3)?

log₃(4/x-3)

300

Expand: log(5x^2)

log(5) + 2log(x)

400

In a logarithmic equation with two solutions, how do I check whether they are extraneous? 

Plug in for x, argument of each log must be > 0.

500

Solve: log(x) + log(x-4) = log(21)

x = 7

Misc.

100

What is the inverse of a logarithmic function?

An exponential function

200

What type of asymptotes do logarithmic graphs have?

Vertical Asymptotes

300

What are the steps to graphing a log function?

1.) Find the inverse

2.) Graph the inverse (exponential)

3.) Switch coordinate points and graph new function (log)

400

How does changing the base of a log function affect the graph?

Lower base means the graph will have a steeper curve

500

What point does every graph of the form y = log_bx, where b > 1, pass through?

(1,0)