Problem 134 - Precalculus

Question

For the following exercises, rewrite each equation in exponential form. $$\log _{y}(137)=x$$

Step-by-Step Solution

Answer

The exponential form is \( y^x = 137 \).

Step 1: Understand the Relationship

The logarithmic equation we are given is \( \log_{y}(137) = x \). In a logarithmic equation, \( \log_{b}(a) = c \) can be rewritten in exponential form as \( b^c = a \). This means the base of the logarithm raised to the power of the result equals the number inside the logarithm.

Step 2: Rewrite in Exponential Form

Using the understanding from Step 1, we need to rewrite the logarithmic form \( \log _{y}(137) = x \) in the form of an exponential equation using the formula \( b^c = a \). Here, \( b = y \), \( a = 137 \), and \( c = x \). Therefore, the exponential form of the equation is \( y^x = 137 \).

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical expressions that involve logarithms. A logarithm essentially answers the question: "To what power must the base be raised to obtain a certain number?" In a logarithmic equation, this is typically represented as \( \log_{b}(a) = c \). Here:

• \( b \) is the base of the logarithm.
• \( a \) is the number you want to find the logarithm of.
• \( c \) is the logarithm itself or the exponent needed.

Such equations help solve problems where you need to determine the exponent required to achieve a certain value from a specific base.
Understanding logarithmic equations is crucial for progressing in various fields such as science, engineering, and financial analysis, where exponential growth or decay is essential.

Exponential Equations

Exponential equations are those in which variables appear as exponents. These are typically given in the form \( b^c = a \), which corresponds directly to a logarithmic format. In the exercise example, the logarithmic equation \( \log_{y}(137) = x \) can be rewritten as the exponential equation \( y^x = 137 \).

• \( b \) or \( y \) is the base.
• \( c \) or \( x \) is the exponent or power.
• \( a \) or \( 137 \) is the result of raising the base to that power.

Understanding exponential equations is essential for modeling phenomena where quantities grow or shrink rapidly, such as population growth, radioactive decay, and financial investments. They provide insight into the relationships between factors that change rapidly over time.

Logarithm Base Change

Sometimes, it is useful to change the base of a logarithm to make calculations easier or to compare logarithms with different bases. This is done using the change of base formula:\[\log_{b}(a) = \frac{\log_{k}(a)}{\log_{k}(b)}\]Here, \( k \) is any positive number you choose (most commonly 10 or \( e \), for natural logarithms). This formula helps when you want to use a calculator that only computes logarithms of certain bases, or when simplifying expressions with different bases.

• The ability to change the base is useful in solving equations across different disciplines.
• It provides flexibility in calculations, allowing for conversions between incompatible systems.

By mastering this concept, students can become more adept at solving complex logarithmic equations and handling various mathematical challenges with ease.