5.3.1 Evaluating Logarithmic Functions

Question 1

Evaluating Logarithmic Functions

Answer 1

• Remember: The change of base theorem, log_bx = log_ax / log_ab, allows you to revise a logarithm problem to be in a base that is easier to use in solving the problem.
• A logarithm indicates the exponent to which you raise a certain base to produce a given value. The inverse of a logarithmic function is an exponential function.
• Logs to the base 10 are written without a base. Logs to the base e are indicated by the symbol “ln.”
• log B (AC) = log B A + log B C
• log B (A/C) = log B A − log B C
• log B (A^C) = C log B A

Question 2

note

Answer 2

• A logarithm is another way of writing an equation that
  involves an exponential term.
• Always remember that a logarithm is an exponent. Whatever the log equals is actually the exponent of the equivalent equation.
• C is the exponent to which you must raise B in order to get A.
  The base of a logarithmic function remains constant.
• Graph a logarithmic function by plotting some points. Notice that for domain values between 0 and 1, this logarithmic curve produces negative range values.
• Logarithmic functions are only defined for positive domain values. The logarithmic function is strictly increasing.
• Remember, a log is an exponent.
• log_B B^A = A is a fancy way of saying, “The exponent to which you must raise B to get B A is A.”
• A log written without a base is assumed to have base 10, which is also called the common log.
• A log with base e is called the natural log and is
  abbreviated “ln.”
• It is a good idea to commit these identities to memory. You can derive them from the definition of a logarithm if you forget them.
• Here are some additional important logarithmic identities.
• The log of a product of two numbers is equal to the sum of the log of the two numbers.
• The log of a quotient of two numbers is equal to the log of the numerator minus the log of the denominator.
• The log of a variable raised to a power is equal to the product of the power and the log of the variable.
• Notice that there are no identities for the log of a sum or for the product of two logs.

Question 3

Evaluate 5log 4 2 + log 4 4

Answer 3

7/2

Question 4

Use a calculator and the change of base formula to evaluate log_7 5.
Change of Base Formula: log_b x=log_n x / log_n b

Answer 4

0.8270875

Question 5

Evaluate 3^(log 3 * 2.714)

Answer 5

2.714

Question 6

What is the domain of the function f(x)=ln(2−x)?

Answer 6

x < 2

Question 7

The rate of a chemical reaction between platinum and carbon dioxide is determined by the constant
k=log_6 9+log_6 4 / log_2 54−log_2 27.
Simplify the expression for k (without using a calculator).

Answer 7

k = 2

Question 8

Evaluate without a calculator.
(log_3 125)(log_5 27)
Hint Use the change of base formula to write the logs with base 10.
Change of Base Formula: log_bx=log_nx/log_nb

Answer 8

None of the above

Question 9

The loudness of a sound in decibels (dB) is given by the equation Loudness=10log(I/I0),where I is the intensity of the sound and I0is a constant equal to 10^−12. If one sound is 5 times as intense as another, how much greater is its loudness in decibels?

Answer 9

7 dB

Question 10

Solve for x:e^x−3e^−x / 2=1

Answer 10

x = ln 3

Question 11

Evaluate without a calculator: log 4 64

Answer 11

3