SECTION 4.4. LOGARITHMIC FUNCTIONS. LOGARITHMIC FUNCTIONS. The logarithm (base b) of a number is the power to which b must be raised to get that number. EXAMPLES:. (a) log 4 16 =. 2. (e) log 5 1 =. 0. (b) log 3 27 =. 3. (f) log 2 1/8 =. - 3. (g) log 10 .1 =. - 1.
The logarithm (base b) of a number is the power to which b must be raised to get that number.
(a) log 4 16 =
(e) log 5 1 =
(b) log 3 27 =
(f) log 2 1/8 =
(g) log10 .1 =
(c) log e e 4 =
(d) log 2 32 =
(h) log 9 27 =
Recall that only one-to-one functions have an inverse.
Exponential functions are one-to-one.
Their inverses are logarithmic functions.
Change the exponential expressions to logarithmic expressions.
1.23 = m
eb = 9
a4 = 24
Change the logarithmic expressions to exponential expressions.
loga4 = 5
loge b = - 3
log3 5 = c
Since the logarithmic function is the inverse of the exponential, the domain of a logarithmic is the same as the range of the exponential.
Find the domain of the functions below:
F(x) = log2 (1 - x)
Logarithm to the base 10.
Ex: log 100 = 2
Logarithm to the base e.
Ex: ln e 2 = 2
Compose the two functions:
g(f(x)) = ln ex = x f(g(x)) = eln x = x
We can see graphically as well as algebraically that these two functions are inverses of each other.
Then f -1(x) = log b x
1. The x-intercept is 1.
2. The y-axis is a vertical asymptote of the graph.
3. A logarithmic function is decreasing if 0 < a < 1 and increasing if a > 1.
4. The graph is continuous.
Graph f(x) = 3log(x – 1). Determine the domain, range, and vertical asymptote of f.
Graph the function f(x) = ln(1 - x).
Determine the domain, range, and vertical asymptote.
Solve: log3(4x – 7) = 2
Solve: logx64 = 2
Solve: e2x = 5
DO EXAMPLE 10 ON ALCOHOL AND DRIVING