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AP Calculus AB: 5.3.1 Evaluating Logarithmic Functions

Mathematics11 CardsCreated 8 months ago

This flashcard set teaches how to evaluate logarithmic expressions using identities and the change of base formula. It emphasizes the relationship between logarithmic and exponential functions, outlines key properties of logs, and includes practical examples using calculators and logarithmic rules.

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Evaluating Logarithmic Functions

  • 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

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Key Terms

Term
Definition

Evaluating Logarithmic Functions

  • 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 ...

note

  • A logarithm is another way of writing an equation that
    involves an exponential term.

  • Always remember that a logarithm is ...

Evaluate 5log 4 2 + log 4 4

7/2

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

0.8270875

Evaluate 3^(log 3 * 2.714)

2.714

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

x < 2

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AP Calculus AB: 5.3.1 Evaluating Logarithmic Functions
TermDefinition

Evaluating Logarithmic Functions

  • 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

note

  • 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.

Evaluate 5log 4 2 + log 4 4

7/2

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

0.8270875

Evaluate 3^(log 3 * 2.714)

2.714

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

x < 2

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).

k = 2

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

None of the above

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?

7 dB

Solve for x:e^x−3e^−x / 2=1

x = ln 3

Evaluate without a calculator: log 4 64

3