Unit 7: Exponential and Logarithmic Functions
Functions that grow by repeated multiplying, the inverse operation that undoes them, and the laws that turn products into sums.
11 topics · 41 questions
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Exponential Functions and Their Graphs
- How it worksHow does the base of an exponential function decide whether its graph grows or decays?
- Worked exampleGraph f(x) = 2x and describe what happens as x gets very large and very negative.
- Common mistakeWhy doesn't an exponential function with base between 0 and 1 count as decreasing to negative values?
- Going furtherWhat happens to the graph of an exponential function when its base gets closer and closer to 1?
Your turn1 of 2
For f(x) = 4x, find f(4).Transformations of Exponential Functions
- How it worksHow do a, h, and k in f(x) = a times b to the power (x - h), plus k reshape the basic exponential graph?
- Worked exampleFind the transformation of f(x) = 2x that produces g(x) = 3 times 2 to the power (x - 1) minus 4.
- Common mistakeWhy doesn't shifting an exponential function left or right change its long-term end behavior?
- Ties togetherHow does shifting and stretching an exponential function preview the a, h, and k rules used on every other function?
Your turn1 of 2
The function f(x) = 4x is shifted right by 8, vertically stretched by a factor of 7, and shifted up by 3. Write g(x).Defining Logarithms as the Inverse of Exponentials
- How it worksHow does log base b of x undo raising b to a power, the same way a square root undoes squaring?
- Worked exampleFind the logarithmic form of 25 = 32 and the exponential form of log base 3 of 81 = 4.
- Common mistakeWhy isn't log base 2 of 8 the same thing as 8 divided by 2?
Your turn1 of 2
Let f(x) = log base 7 of x. Find its inverse function f−1(x).Graphing Logarithmic Functions
- How it worksHow does reflecting the graph of y = 2x across the line y = x produce the graph of y = log base 2 of x?
- Worked exampleGraph y = log base 2 of x and identify its vertical asymptote and domain.
- Ties togetherHow does the domain of a logarithmic function connect to the range of the exponential function it inverts?
Your turn1 of 2
Find the vertical asymptote (the x-value that bounds the domain) of y = log base 7 of (x - 6).The Product, Quotient, and Power Laws of Logarithms
- How it worksHow does the exponent rule for multiplying powers with the same base turn into the product law of logarithms?
- Worked exampleFind the expanded form of log base 5 of (25x3 over y) using the properties of logarithms.
- Common mistakeWhy isn't log(a+b) equal to log a plus log b?
- Common mistakeWhy doesn't log base b of x to the n equal log base b of x, raised to the n?
Your turn1 of 2
Find the expanded form of log(103·x4 / y), using the properties of logarithms.Condensing and Expanding Logarithmic Expressions
- How it worksHow do the product, quotient, and power laws combine to condense three log terms into one?
- Worked exampleFind the single logarithm equal to 2 log x plus log y minus 3 log z.
- Common mistakeWhy can't you condense log x plus log y plus log z into log of (x + y + z)?
- Going furtherWhat changes about condensing a log expression when one term has a fractional coefficient, like one half times log x?
Your turn1 of 2
Find the single logarithm equal to 3 log(x) + log(y) - 3 log(z).Common and Natural Logarithms
- How it worksHow does the natural logarithm ln x relate to log base e of x on a calculator?
- Worked exampleFind the value of ln(e3) and log(1000) without a calculator.
- Common mistakeWhy doesn't log on a calculator with no base written mean log base e?
Your turn1 of 2
Find the value of ln(e4) plus log(104).The Change of Base Formula
- How it worksHow does dividing two logs of a different, known base recompute a log in a base your calculator doesn't have?
- Worked exampleFind the value of log base 7 of 50 using the change of base formula.
- Common mistakeWhy doesn't the change of base formula care which new base you switch to?
- Ties togetherHow does the change of base formula let you graph a logarithm of any base using the same reflection trick as base 2?
Your turn1 of 2
Using the change of base formula, write log base 7 of x, using base-10 logs, as a function of x.Solving Exponential Equations with Logarithms
- How it worksHow does taking the log of both sides bring a variable exponent down where you can solve for it?
- Worked exampleSolve 5x + 1 = 40 for x, giving an exact answer in terms of logarithms.
- Common mistakeWhy can't you solve 5x+1 = 40 by just dividing both sides by 5?
Your turn1 of 2
Solve 7x + 6 = 932 for x, giving an exact answer in terms of logarithms.Solving Logarithmic Equations
- How it worksHow does rewriting a logarithmic equation in exponential form clear the logs away entirely?
- Worked exampleSolve log base 2 of (x + 3) = 4 for x.
- Common mistakeWhy does a logarithmic equation sometimes produce a solution you have to throw out for making the log's input negative?
- Common mistakeWhy can adding two logarithms together before condensing them still leave you with an extraneous solution afterward?
- Going furtherWhat has to be checked before accepting a solution to an equation with two different logarithms in it?
Your turn1 of 2
Solve log base 4 of (x + 8) = 3 for x.Logarithmic Number Lines and Scales
- How it worksHow does spacing numbers by their logarithm turn multiplication on the original scale into addition on the new one?
- Worked exampleShow me the numbers 1, 10, 100, and 1000 placed on a logarithmic number line and how they space out.
- Common mistakeWhy doesn't doubling a value move it the same distance every time on a logarithmic scale?
- Going furtherWhat happens to the spacing on a logarithmic scale when you compare it to a linear number line for the same numbers?
Your turn1 of 2
On a logarithmic number line built from powers of 7, how many equally-spaced steps separate 73 from 79?