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Chapter 4: Problem 120

Describe the relationship between an equation in logarithmic form and an equivalent equation in exponential form.

Short Answer

Expert verified
Logarithmic and exponential forms of equations are interchangeable. For an equation in exponential form \(a = b^c\), its equivalent logarithmic form would be \(\log_b a = c\). Conversely, for a logarithmic equation \(\log_b a = c\), its equivalent exponential form would be \(a = b^c\).

Step by step solution

01

Defining the Forms

Firstly, understand the equation in its exponential form: \(a = b^c\) where a is the result of b (the base) raised to the power c (the exponent). 'b' is always a positive number other than 1, 'a' is positive and 'c' can be any real number. An equivalent logarithmic form of this equation is written as \(\log_b a = c\) in which \(\log_b a\) denotes the number you would need to raise 'b' to get 'a'.
02

Converting Exponential to Logarithmic form

If we have an exponential equation \(a = b^c\), it could be converted to logarithmic form as follows: Start by identifying 'a', 'b' and 'c' from the exponential form. 'b' is the base of the logarithm, 'a' is the number we're taking the log of, and 'c' is the value the logarithm should equal. Hence we can write this as a logarithmic equation \(\log_b a = c\)
03

Converting Logarithmic to Exponential form

The process of conversion from logarithmic to exponential form is just the reverse of the above process. If we have a logarithmic equation \(\log_b a = c\), 'b' is the base raised to equate 'c' to give 'a'. Thus the equation becomes \(a = b^c\) in exponential form

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Most popular questions from this chapter

Evaluate or simplify each expression without using a calculator. $$ \ln e^{9 x} $$

Will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$\log _{3} 81, \text { or } \log _{3} 9^{2} ?$$

The figure shows the graph of \(f(x)=\ln x\). Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ g(x)=1-\ln x $$

In Exercises \(125-128,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \frac{\log _{7} 49}{\log _{7} 7}=\log _{7} 49-\log _{7} 7 $$

The loudness level of a sound, \(D,\) in decibels, is given by the formula $$D=10 \log \left(10^{12} I\right)$$ where \(I\) is the intensity of the sound, in watts per meter \(^{2}\). Decibel levels range from \(0,\) a barely audible sound, to \(160,\) a sound resulting in a ruptured eardrum. (Any exposure to sounds of 130 decibels or higher puts a person at immediate risk for hearing damage.) Use the formula to solve. The sound of a blue whale can be heard 500 miles away, reaching an intensity of \(6.3 \times 10^{6}\) watts per meter\(^{2}\). Determine the decibel level of this sound. At close range, can the sound of a blue whale rupture the human eardrum?
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