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Chapter 4: Problem 134

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

Short Answer

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

Step by step solution

01

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.
02

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 \( y^x = 137 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

For the following exercises, refer to Table 4.29. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} \\ \hline f(x) & {7.5} & {6} & {5.2} & {4.3} & {3.9} & {3.1} & {2.9} \\ \hline\end{array}$$ Use the LOGarithm option of the REGression feature to find a logarithmic function of the form \(y=a+b \ln (x)\) that best fits the data in the table.

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth. $$ \ln (15) $$

For the following exercises, condense each expression to a single logaritim using the properties of logaritims. $$ \ln \left(6 x^{9}\right)-\ln \left(3 x^{2}\right) $$

For the following exercises, refer to Table 4.30. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} & {10} \\ \hline f(x) & {8.7} & {12.3} & {15.4} & {18.5} & {20.7} & {22.5} & {23.3} & {24} & {24.6} & {24.8} \\\ \hline\end{array}$$ Use a graphing calculator to create a scatter diagram of the data.

The temperature of an object in degrees Fahrenheit after \(t\) minutes is represented by the equation \(T(t)=68 e^{-0.017 t_{t}}+72 .\) To the nearee, what is the temperature of the object after one and a half hours?
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