Problem 11 - College Algebra

Question

For the following exercises, rewrite each equation in exponential form. $$\log _{15}(a)=b$$

Step-by-Step Solution

Answer

The exponential form is \(15^{b} = a\).

Step 1: Identify the logarithmic equation

The equation given is \( \log_{15}(a) = b \). This is a logarithmic equation where 15 is the base, \( a \) is the argument, and \( b \) is the result.

Step 2: Convert the equation to exponential form

To convert the logarithmic equation \( \log_{15}(a) = b \) to exponential form, we use the rule that states \( \log_{b}(x) = y \) implies \( b^{y} = x \). Applying this to \( \log_{15}(a) = b \), we get \( 15^{b} = a \).

Key Concepts

Logarithmic Equations

Logarithmic equations can sometimes appear daunting at first glance. But they are simply mathematical statements that express the relationship between three numbers. In a logarithmic equation like \( \log_{b}(x) = y \), the equation represents: The power \( y \) that the base \( b \) must be raised to produce the number \( x \).
Here's the breakdown:

• "\( b \)" is the base of the logarithm.
• "\( x \)" is the argument, sometimes called the antilogarithm.
• "\( y \)" is the result of the logarithmic operation.

When you get a logarithmic equation like \( \log_{15}(a) = b \), it tells you the power to which the base 15 must be raised to yield \( a \). Look at it like a puzzle — the equation is just waiting to show you its secrets when you know how to unlock them.

Base and Argument

The terms 'base' and 'argument' are fundamental in understanding both logarithmic and exponential forms.

• The **base** in a logarithm, denoted here as 15 in \( \log_{15}(a) = b \), is the number that is repeatedly multiplied.
• The **argument**, which is \( a \) in this example, is the number you are taking the logarithm of.

Knowing these components helps in deciphering the equation. The base is like the root structure of a tree, supporting and determining the outcome or the height (argument) when raised to a certain power (the result of the logarithm). By practicing identifying the base and argument, you become more comfortable converting and manipulating logarithmic and exponential equations easily.

Conversion of Logarithmic to Exponential Form

Converting a logarithmic equation to its exponential form is a handy skill that makes solving these equations much easier.
To comprehend the conversion, remember this guiding principle:
- If \( \log_{b}(x) = y \), then this can be rewritten in exponential form as \( b^{y} = x \).
In our example, \( \log_{15}(a) = b \) becomes \( 15^{b} = a \). What this means is you are shifting perspective:

• Instead of asking, "What power of 15 gives \( a \)?" which is the logarithmic statement,
• you say, "15 to the power of \( b \) equals \( a \)," which is the exponential form.

This transformation assists in solving problems where manipulating or isolating variables is required. It's like translating and thereby simplifying the language of mathematics, ensuring clarity in understanding equations.