MTH 161 Ch3 Log Properties and Equations (3.4-3.6)

Product rule for logs?

log base b of (MN) = log base b of M plus log base b of N.

Quotient rule for logs?

log base b of (M/N) = log base b of M minus log base b of N.

Power rule for logs?

log base b of (M^p) = p times log base b of M.

What is log of a sum, log(M + N)?

It does NOT split. There is no rule for log of a sum or difference.

Change of base formula?

log base b of x = ln x / ln b (or log x / log b).

Expand log base 2 of (8x^3 / y).

3 + 3 log base 2 of x minus log base 2 of y.

Condense 2 ln x + ln y.

ln (x^2 y).

How do you solve an exponential equation with the same base?

Rewrite both sides with the same base, then set the exponents equal. Example: 2^x = 32 gives x = 5.

How do you solve an exponential equation with different bases?

Isolate the exponential, take the log or ln of both sides, use the power rule to bring down the exponent, then solve.

Solve 3^x = 20.

x = ln 20 / ln 3, about 2.73.

How do you solve a log equation?

Combine logs into one, rewrite in exponential form, solve, and check the answers in the original equation.

Why must you check log equation solutions?

A solution that makes any log argument zero or negative is extraneous and must be rejected.

Solve log base 2 of x = 5.

x = 2^5 = 32.

Solve log (x) + log (x - 3) = 1.

log (x(x-3)) = 1, so x^2 - 3x = 10, giving x = 5 or x = -2. Reject -2. Answer x = 5.

Compound interest formula?

A = P(1 + r/n)^(nt), where P is principal, r the annual rate, n compounds per year, t years.

Continuous compounding formula?

A = P e^(rt).

Exponential growth and decay model?

A = A0 e^(kt). k greater than 0 is growth, k less than 0 is decay.

How do you find doubling time?

Solve 2 = e^(kt), so t = ln 2 / k.

What is half-life?

The time for a decaying quantity to fall to half. Solve 1/2 = e^(kt) for t.

How do you build a model from two data points?

Plug each point into A = A0 e^(kt) to get A0 and k, then use the model to answer questions.