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Chapter 3: Problem 20

write each equation in its equivalent logarithmic form. $$8^{y}=300$$

Short Answer

Expert verified
The equivalent logarithmic form of the equation \(8^{y} = 300\) is \(\log_{8} 300 = y\).

Step by step solution

01

Understand the relationship between logs and exponents

Consider that every exponential equation \(b^{x} = y\), where 'b' is the base, 'x' is the exponent, and 'y' is the result, can be rewritten in logarithmic form as \(\log_{b} y = x\), where \(\log_{b} y\) is the log base 'b' of 'y', and 'x' is the number for which 'b to the power of x' equals 'y'.
02

Apply the rule to the given equation

Rewrite the equation \(8^{y} = 300\) in logarithmic form using the conversion rule outlined in Step 1.
03

Putting it all together

Applying the aforementioned rule, the equation now becomes \(\log_{8} 300 = y\). This is the logarithmic equivalent of the original exponential equation.

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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 variable appears in the exponent. In these equations, one deals with quantities that grow by a constant factor over equal intervals. Commonly, exponential equations take a form like this:
  • \( b^x = y \)
Here, \( b \) is the base, \( x \) is the exponent or the power, and \( y \) is the result. These are fundamental in describing phenomena like population growth, radioactive decay, and compound interest. Since they involve a power (and not a linear or polynomial form), solving them often requires using logarithms, especially when trying to isolate variables that are exponents.
Logarithmic Form
The logarithmic form is a way of expressing an equation involving a logarithm. It is particularly useful when solving exponential equations where the unknown is in the exponent. Converting an exponential equation to logarithmic form can help us isolate and solve for variables. The rule for conversion is:
  • From \( b^x = y \) to \( \log_b y = x \)
This conversion demonstrates the concept that the logarithm of a number, in a given base, tells us to what power the base must be raised to obtain that number. For example, converting \( 8^y = 300 \) into logarithmic form, we get \( \log_8 300 = y \). This form makes it easier to solve for \( y \) if needed since it may allow the use of calculators or logarithmic tables.
Conversion Between Logarithms and Exponents
Conversion between logarithms and exponents is an essential skill for solving many types of mathematical problems. The process revolves around understanding the equivalence between exponential and logarithmic forms, especially when dealing with exponential growth or decay. Key points to remember include:
  • An exponential form \( b^x = y \) can be rewritten as a logarithmic form \( \log_b y = x \).
  • This conversion allows us to find the exponent \( x \) when given the base and the result, or find the result \( y \) given the base and the exponent.
  • Being comfortable with moving between these forms helps solve equations that may not be easily solvable with direct arithmetic methods.
Understanding this conversion helps us bridge the gap between exponential expressions and their logarithmic counterparts, making complex equations more manageable, and is a cornerstone in tackling real-world problems involving exponential changes.

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Most popular questions from this chapter

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\log _{6}\left(\frac{x-1}{x^{2}+4}\right)=\log _{6}(x-1)-\log _{6}\left(x^{2}+4\right)$$

Explain why the logarithm of 1 with base \(b\) is 0.

graph f and g in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of f. $$f(x)=\ln x, g(x)=\ln (x+3)$$

In many states, a \(17 \%\) risk of a car accident with a blood alcohol concentration of 0.08 is the lowest level for charging a motorist with driving under the influence. Do you agree with the \(17 \%\) risk as a cutoff percentage, or do you feel that the percentage should be lower or higher? Explain your answer. What blood alcohol concentration corresponds to what you believe is an appropriate percentage?

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\log _{b} x\) is the exponent to which \(b\) must be raised to obtain \(x\)
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