Question

Express the following statements using logarithms: (a) \(8^{2}=64\) (b) \(4^{3}=64\) (c) \(2^{6}=64\)

Short answer

Question: Rewrite the given exponential equations in logarithmic form: (a) \(8^{2}=64\) (b) \(4^{3}=64\) (c) \(2^{6}=64\) Answer: (a) \(\log_{8}(64)=2\) (b) \(\log_{4}(64)=3\) (c) \(\log_{2}(64)=6\)

Step-by-step solution

(Step 1: Understand the logarithmic function)

The logarithm function is the inverse of an exponential function. Given an equation \(a^x=b\), we can rewrite it as a logarithm equation using the following format: \(\log_{a}(b)=x\). In this problem, we have three equations to transform from the exponential form to the logarithmic form.

(Step 2: Transform equation a)

For the first equation \(8^{2}=64\), we can rewrite it using the logarithm with base 8 as follows: \(\log_{8}(64)=2\).

(Step 3: Transform equation b)

For the second equation \(4^{3}=64\), we can rewrite it using the logarithm with base 4 as follows: \(\log_{4}(64)=3\).

(Step 4: Transform equation c)

For the third equation \(2^{6}=64\), we can rewrite it using the logarithm with base 2 as follows: \(\log_{2}(64)=6\). In conclusion, the given statements have been expressed in logarithmic form as: (a) \(\log_{8}(64)=2\) (b) \(\log_{4}(64)=3\) (c) \(\log_{2}(64)=6\)

Key concepts

Exponential Functions

An exponential function is a mathematical expression in which a variable appears in the exponent. A simple form of an exponential function is given by \( y = a^x \), where \( a \) is the base and \( x \) is the exponent. This type of function shows a rapid increase or decrease due to the exponentiation operation.
Exponential functions are powerful tools in mathematics because they model a wide variety of real-world situations, such as compound interest, population growth, and radioactive decay.
When examining exponential equations like the ones in the exercise, you often find equations in the form \[ a^x = b \].
This form can be tricky to solve directly when \( x \) is unknown, prompting the need for logarithmic functions.

Inverse Functions

Inverse functions are functions that "undo" each other. For example, addition and subtraction are inverse operations; they reverse each other's effects. Similarly, exponential and logarithmic functions are inverse functions.

• In mathematical notation for a base \( a \), if \( y = a^x \), then the inverse function is \( x = \log_{a}(y) \).
• This means that if you raise a number \( a \) to a certain power \( x \), you can use the logarithm to get back to \( x \) from \( y \).
• This property is particularly useful in solving certain types of equations, especially those where the variable is an exponent.

Understanding inverse functions helps simplify complex problems and understand processes that have exponential growth or decay.

Logarithmic Form

The logarithmic form is a special way of writing equations that allows you to solve for unknown exponents. It transforms an exponential equation into a logarithmic one. The general form of a logarithm is \( \log_{a}(b) = x \), and it is read as "log base \( a \) of \( b \) equals \( x \)."

• This relation indicates "to what power must \( a \) be raised, to produce \( b \)?"
• In our exercise, each exponential equation \( 8^2=64 \), \( 4^3=64 \), and \( 2^6=64 \) is successfully rewritten using logarithms: \( \log_{8}(64)=2 \), \( \log_{4}(64)=3 \), and \( \log_{2}(64)=6 \).
• Identifying and writing equations in logarithmic form simplifies complex calculations, especially those involving exponential changes.

Converting to logarithmic form not only helps in solving equations but also provides a deeper understanding of the operations underlying exponential growth.