Logarithms

Logarithm

A logarithm answers the question: 'What exponent do we need to raise the base to, in order to get this number?'

Exponential functions

Functions like eˣ that are considered inverses of logarithmic functions.

Logarithmic functions

Functions like ln(x) that are considered inverses of exponential functions.

Key identities of logarithms and exponentials

e^(ln(x)) = x for x > 0 and ln(eˣ) = x for all real x.

Restriction on logarithmic function input

The input x must be greater than 0 (x > 0) because the output of any exponential function b^exponent is always positive.

logb(M) + logb(N)

log*b(MN)

ln(5) + ln(4)

ln(20)

logb(M) - logb(N)

log*b(M/N)

log(20) - log(2)

log(10) = 1

C * log_b(M)

log_b(M^C)

3 log_2(5)

log_2(5^3) = log_2(125)

log_a(b)

\frac{\logc(b)}{\logc(a)}

Usefulness of Change of Base Formula

It allows you to compute a logarithm with any base using a calculator, which typically only has buttons for log (base 10) and ln (base e).

Equation with single logarithm

You have an equation with a single logarithm set equal to a number, like ln(2x) = 5.

What is Step 1 for solving an equation with a single logarithm like ln(2x) = 5?

Isolate the logarithm.

What is Step 2 for solving an equation with a single logarithm like ln(2x) = 5?

Rewrite the equation in exponential form.

What is the final, crucial step when solving an equation with a single logarithm?

Check solution(s) against the domain of the original logarithmic expression, ensuring the argument is positive ( > 0).

Equation with logarithm on both sides

You have an equation with a logarithm on both sides with the same base, like log4(x-1) = log4(8).

When solving an equation with logarithms on both sides with the same base, like log4(x-1) = log4(8), what can you do immediately and why?

You can set the arguments equal to each other (x-1 = 8) because logarithmic functions are one-to-one.

What is the final, crucial step when solving an equation with logarithms on both sides?

Check solution(s) against the domain of the original logarithmic expression(s), ensuring the argument(s) are positive ( > 0).

Equation with multiple logarithms

You have an equation with multiple logarithms on one side, like log(x) + log(x+1) = 1.

When solving an equation with multiple logarithms on one side, like log(x) + log(x+1) = 1, what must you do before you can solve it?

Condense the multiple logarithms into a single logarithm using the logarithm properties.

What is the final, crucial step after solving an equation with multiple logarithms?

Check solution(s) against the domain of all original logarithmic expressions, ensuring all arguments are positive ( > 0).

Handling multiple solutions in log equations

You solve a log equation and get two answers: x = 5 and x = -2.

When solving a log equation and you get solutions like x = 5 and x = -2, what do you do with x = -2 and why?

Discard x = -2 because the argument of a logarithm must be positive. Plugging -2 back into an original term like ln(x) would result in ln(-2), which is undefined.