05/10 Logarithmic Functions Free Online FlashCards

What does log_b(x)=y mean?

A logarithm gives the exponent needed to produce a number: log_b(x)=y means b^y=x, where b>0, b≠1, and x>0.

What is the domain condition for log_b(x)?

The argument must be positive, so log_b(x) is defined only for x>0.

What is the inverse of f(x)=b^x?

The exponential function f(x)=b^x and the logarithmic function f⁻¹(x)=log_b(x) are inverses.

State the product property of logarithms.

log_b(MN)=log_b(M)+log_b(N), for positive M and N.

State the quotient property of logarithms.

log_b(M/N)=log_b(M)−log_b(N), for positive M and N.

State the power property of logarithms.

log_b(M^p)=p log_b(M). The exponent becomes a coefficient.

What is the natural-log change-of-base formula?

log_b(M)=ln(M)/ln(b), provided M>0, b>0, and b≠1.

What are key features of y=log_b(x)?

The graph has domain (0,∞), range (−∞,∞), and vertical asymptote x=0.

How do h and k affect f(x)=a log_b(x−h)+k?

For f(x)=a log_b(x−h)+k, the vertical asymptote is x=h and the domain is x>h.

Solve log_4(x)=3.

Convert to exponential form: 4^3=x, so x=64.

Solve log_3(2x−1)=log_3(11).

Set the arguments equal: 2x−1=11. Therefore, x=6, which satisfies the domain condition.

Solve 2^x=7.

x=ln(7)/ln(2)≈2.807. Take natural logarithms of both sides and use the power property.