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Chapter 3: Problem 15

Write each equation in its equivalent logarithmic form. $$13^{2}=x$$

Short Answer

Expert verified
The equivalent logarithmic form of the given equation \(13^2 = x\) is \(\log_{13} x = 2\).

Step by step solution

01

Identify the Base, Exponent and Result

In the given equation \(13^2 = x\), the base is \(13\), the exponent is \(2\), and the result is \(x\).
02

Convert to Logarithmic Form

In the equivalent logarithmic form, the equation will become \(\log_{13} x = 2\). This is achieved by following the relationship between the exponential and logarithmic forms: \(\log_a c = b\). In this case, \(a\) is \(13\), \(c\) is \(x\), and \(b\) is \(2\).

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Key Concepts

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

Exponential Equations
Exponential equations are mathematical expressions where a number, known as the base, is raised to a power, known as the exponent. This typically takes the form of \(a^b = c\), where \(a\) is the base, \(b\) is the exponent, and \(c\) is the result.

Understanding how to solve these equations is crucial for working with logarithmic forms. These types of expressions are used in various fields including science and engineering to model growth, decay, and other phenomena.

When dealing with exponential equations, the goal is often to find either the exponent or the base when the result is given. This leads to an understanding of the inverse operation, which is the logarithm.
Base and Exponent
In the world of exponential equations, the base is the number that is being multiplied by itself, while the exponent indicates how many times the base is used in the multiplication. For example, in the equation \(13^2\), 13 is the base and 2 is the exponent.

Here's a closer look at what they mean:
  • Base: In the equation \(13^2 = x\), 13 is the base. It is the number that you're repeatedly multiplying.
  • Exponent: The 2 in \(13^2\) is the exponent. It tells you how many times the base is used in multiplication. So, 13 is multiplied by itself once.
Understanding these components is essential for comprehending other mathematical concepts, such as converting between exponential and logarithmic forms.
Conversion Between Forms
Converting between exponential and logarithmic forms is an important skill in mathematics. It allows us to express equations in different ways, which can be more useful depending on the problem at hand.

You can convert an exponential equation, like \(13^2 = x\), into its logarithmic form using the relationship: \(\log_a b = c\). For our example, this means \(\log_{13} x = 2\).

Here's a simple guide on how to do it:
  • Identify the base, exponent, and result in the exponential form.
  • Write the logarithm with the base as its subscript to the log function \(\log_{base}(result) = exponent\).
By mastering these conversions, you improve your ability to solve a wider array of problems and understand more complex mathematical relationships. Logarithms can simplify multiplication processes into addition, making them extremely useful for calculations.

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Most popular questions from this chapter

Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use \(3^{x}=140\) in your explanation.

Describe the shape of a scatter plot that suggests modeling the data with an exponential function.

Solve and graph the solution set on a number line: \(2 x^{2}+5 x<12 .\)

In Exercises \(125-132,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation.. $$3^{x}=2 x+3$$

Nigeria has a growth rate of 0.025 or \(2.5 \% .\) Describe what this means.
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