Name:
Logarithms Name: _____________________________period:____
A LOGARITHM IS AN EXPONENT! This means that logax is the exponent to which a must be raised to obtain x. For instance, log28 = 3 because 23 = 8. Another example? log416 = 2 because 42 = 16.
Evaluate the given expression without a calculator:
1) log2 16 = _________ 2) log5 [pic] = ___________ 3) log16 4 = __________
4) log10 0.01 = __________ 5) log10 1000 = __________ 6) log3 0 = __________
Since logax is the inverse function of ax, it follows that the domain of logax is the range of ax, (0, [pic]). In other words, logax is defined only if x is positive.
7) On the coordinate plane to the right, sketch the graphs of f(x) = 2x and g(x) = log2x. What do you notice?
Properties of logarithms:
**we will focus on these ** loga1 = 0 (because a0 = 1) loga(uv) = logau + logav
** logaa = 1 (because a1 = a) loga[pic] = logau - logav
** logaax = x (because ax = ax) ** logaun = n(logau)
The most widely used base for logarithms is the number e. The logarithmic function with base e is called the natural logarithmic function and is denoted by the special symbol lnx.
Important! ( logex = lnx, x > 0.
Properties of natural logs: ** ln1 = 0 ln(uv) = lnu + lnv
** lne = 1 ln[pic] = lnu - lnv
** lnex = x ** lnun = n(lnu)
Evaluate the given expression without a calculator:
10) lne3 = __________ 11) ln[pic] = __________ 12) lne-2 = __________
13) ln 1 = __________ 14) logaa2 = __________ 15) loga[pic] = _________
Solve for x. Use the definition of a log to write the equation in log form:
16) 2x = 8 ________ 20) 53 = 125 __________________
17) 10x = 0.1 ________ 21) 811/4 = 3 __________________
18) 4x = 1 ________ 22) e3 = 20.0855… __________________
19) ex = e ________ 23) ex = 4 __________________
Solving: If you have 2x = 7, natural log both sides!
ln2x = ln7
xln2 = ln7
x = [pic] [pic] 2.81.
Solve for x:
31) 4e2x = 5 33) lnx = 3
32) 4x + 3 = 7x 34) 2ln3x = 4
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