Logarithmic Functions

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Logarithmic functions, or logs, are the inverse of exponential functions.  The two can be written in either form. 

y = b^x     can be rewritten as    log_b(y) = x 

The main thing to remember about log functions is that they produce exponents.

EXAMPLES

Evaluate log₂ 32.

The question here is basically, "What exponent of 2 will give you 32?"  Remembering that 2⁵ = 32, it follows that log₂ 32 = 5.

Evaluate log₃ 9.

Don't be fooled here.  This is not a division problem; the question is asking for an exponent.  Since 3² = 9, this means log₃ 9 = 2.

Evaluate log₅ (¹/₂₅).

Remembering that negative exponents make reciprocals, we find that 5^-2 = ¹/₂₅.  So log₅ (¹/₂₅) = -2.

Evaluate log₇ (-49).

There is no such exponent that satisfies 7^? = -49.  Therefore, log₇ (-49) does not exist.

Using the definition of logarithm, you can also solve equations.

EXAMPLE

Solve:    log₂(x) = 4

Understand:

This reads "the log base two of a number equals four."  Solve by converting the logarithmic statement into its equivalent exponential form.

Solution:

log₂(x) = 4

2⁴ = x

16 = x 

You can also get the graph of a log function from the graph of an exponential function.

Recall how y = 2^x looks on the graph:

The domain is all real numbers, the range is only the positive numbers, there's a y-intercept at 1, and there's a horizontal asymptote through y = 0.

If we think about reflecting this graph over the diagonal line y = x, we get the graph of y = log₂(x).

Notice with this function, the domain is only the positive numbers (remember how log₇ (-49) didn't exist?), the range is all real numbers, there's an x-intercept at 1, and there's a vertical asymptote through x = 0.

Like with exponential functions, the appearance of the log function graph varies depending on the base, but all log graphs have the same general shape:

y = log₂ (x)

y = log₃ (x)

y = log₅ (x)

y = log₁₀ (x)

The base 10 logarithm is also known as the common logarithm.  If the base isn't written, it's understood to be 10.

log (x) = log₁₀ (x)

Practice

1.  Evaluate the following, if possible:  a) log₄ 64  b) log₂ (−16)  c) log₃ (¹/₂₇) 

2.  Evaluate the following, if possible:  a) log₉ 3  b) log₇ 1  c) log 100

Answers

1.  a) 3  b) not possible  c) −3

2.  a) ¹/₂  b) 0  c) 2