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Chapter 4: Problem 13

Change to exponential form. (a) \(\log x=50\) (b) \(\log x=20 t\) (c) \(\ln x=0.1\) (d) \( \ln w=4+3 x\) (e) \(\ln (z-2)=\frac{1}{6}\)

Short Answer

Expert verified
(a) \( x = 10^{50} \), (b) \( x = 10^{20t} \), (c) \( x = e^{0.1} \), (d) \( w = e^{4+3x} \), (e) \( z-2 = e^{\frac{1}{6}} \).

Step by step solution

01

Understanding Logarithmic and Exponential Forms

Before we rewrite each expression in exponential form, it's important to remember that if you have a logarithmic equation in the form \( \log_b(x) = y \), it can be expressed in exponential form as \( x = b^y \). When dealing with natural logarithms (\( \ln \)), the base is \( e \).
02

Change \( \log x = 50 \) to Exponential Form

The equation \( \log x = 50 \) is assumed to be in common log form, which means the base is 10. Thus, the exponential form is \( x = 10^{50} \).
03

Change \( \log x = 20t \) to Exponential Form

Similarly, \( \log x = 20t \) implies the base is 10. In exponential form, this is \( x = 10^{20t} \).
04

Change \( \ln x = 0.1 \) to Exponential Form

For the natural log, \( \ln x = 0.1 \), convert to exponential form using the base \( e \): \( x = e^{0.1} \).
05

Change \( \ln w = 4 + 3x \) to Exponential Form

Use the base \( e \) for\( \ln w = 4 + 3x \), resulting in \( w = e^{4 + 3x} \).
06

Change \( \ln (z-2) = \frac{1}{6} \) to Exponential Form

Convert \( \ln (z-2) = \frac{1}{6} \) to exponential form using base \( e \): \( z - 2 = e^{\frac{1}{6}} \).

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Key Concepts

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

Logarithmic Form
Logarithmic form is a way of representing an equation that involves the logarithm of a number. It is represented as \( \log_b(x) = y \), where \( b \) is the base of the logarithm, \( x \) is the number, and \( y \) is the result of the logarithm. From this form, you can easily convert the equation into its exponential form, which is \( x = b^y \). Converting between logarithmic and exponential forms is essential in mathematics because it helps in solving equations more easily.
To better understand, consider these key points:
  • The base \( b \) must always be a positive number and cannot be 1.
  • The logarithm represents the power to which the base must be raised to achieve that specific number \( x \).
  • Logarithmic equations simplify multiplication into addition, division into subtraction, and powers into multiplication.
Understanding this conversion process is crucial for transforming and solving logarithmic equations, whether they are in common or natural logarithm forms.
Natural Logarithm
The natural logarithm, often denoted as \( \ln(x) \), uses the base \( e \). The constant \( e \) is approximately equal to 2.71828, and it's a fundamental constant in mathematics, especially in calculus and exponential growth scenarios.
Some important aspects of natural logarithms are:
  • They arise naturally in problems concerning growth and decay, like calculating compound interest or population growth.
  • The base \( e \) is considered more natural because of the function \( e^x \) being its own derivative, which means its slope at any given point is equal to its value at that point.
  • When converting a natural logarithm from logarithmic to exponential form, you use \( e \) as the base, such as \( \ln x = y \) converts to \( x = e^y \).
Natural logarithms are essential tools in calculus and are used extensively because they make many formulas simpler and more intuitive.
Common Logarithm
Common logarithms are logarithms with base 10 and are often written without a base, as simply \( \log x \). The base-10 logarithm is used mainly because of its simplicity in real-world calculations and its application in scientific notation, where quantities are arranged in powers of ten.
Key points about common logarithms:
  • They are used frequently in calculations involving large numbers, such as in science and engineering.
  • Recognized by \( \log \), whenever the base isn't specified, it is implicitly understood to be 10.
  • Common logarithms help simplify calculations because adding log values corresponds to multiplying the original numbers, and similarly, subtracting corresponds to division.
Converting common logarithms from their logarithmic form to exponential form makes equations easier to work with, particularly when dealing with powers of ten.
Base of Logarithm
The base of a logarithm is the number that is raised to a power in its exponential form. In the logarithmic equation \( \log_b(x) = y \), \( b \) is the base. The base is crucial because it defines the "scaling factor" for the logarithm, indicating how many times \( b \) must be multiplied by itself to reach \( x \).
Important insights into bases of logarithms:
  • The base \( b \) can be any positive number, except for 1.
  • Logarithms with different bases can be converted to each other using the change of base formula: \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \).
  • Common bases include 10 (common logarithm) and \( e \) (natural logarithm), due to their prevalence in practical and theoretical problems.
Understanding the base of logarithms is fundamental for converting between various logarithmic forms and solving logarithmic equations.

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Most popular questions from this chapter

Studies relating serum cholesterol level to coronary heart disease suggest that a risk factor is the ratio \(x\) of the total amount \(C\) of cholesterol in the blood to the amount \(H\) of high-density lipoprotein cholesterol in the blood. For a female, the lifetime risk \(R\) of having a heart attack can be approximated by the formula $$R=2.07 \ln x-2.04 \quad \text { provided } \quad 0 \leq R \leq 1$$ For example, if \(R=0.65,\) then there is a \(65 \%\) chance that a woman will have a heart attack over an average lifetime. (a) Calculate \(R\) for a female with \(C=242\) and \(H=78\) (b) Graphically estimate \(x\) when the risk is \(75 \%\)

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Change \(f(x)=1000(1.05)^{x}\) to an exponential function with base \(e\) and approximate the growth rate of \(f\).
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