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Chapter 11: Problem 15

Rewrite the equation using logarithms instead of exponents. $$ 10^{a^{2} b}=97 $$

Short Answer

Expert verified
Question: Rewrite the equation $$10^{a^{2}b} = 97$$ using logarithms instead of exponents. Answer: $$a^{2}b = \log_{10}{97}$$

Step by step solution

01

Determine the base, exponent, and result for the equation

In the given equation, $$10^{a^{2}b} = 97$$, the base is 10, the exponent is \(a^{2}b\), and the result is 97.
02

Use logarithm property

Use the logarithm property \(b^y = x\) is equivalent to \(y = \log_b{x}\). In this case, \(b = 10\), \(y = a^{2}b\), and \(x = 97\). Using this property, we rewrite the equation as: $$a^{2}b = \log_{10}{97}$$.
03

Write the final logarithmic equation

The final logarithmic equation is: $$a^{2}b = \log_{10}{97}$$.

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Key Concepts

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

Exponential Form
The exponential form of an equation is a way of expressing repeated multiplication. It allows us to write long multiplication processes succinctly. For example, when you see something like \( 10^{a^{2} b} \), it means that the base number, 10, is multiplied by itself to the power of \( a^{2} b \). In simpler terms, if \( a^{2} b \) were just 2, it would mean multiplying 10 twice, or \(10 \times 10\).
Exponential forms are used frequently in math because they provide a clearer way to understand large numbers and complex multiplications.
  • **Base**: The number that is used as a factor in multiplication, here it is 10.
  • **Exponent**: Tells how many times to use the base in multiplication, here it is \( a^{2} b \).
Being comfortable with exponential forms can greatly enhance your number manipulations and insights into algebraic equations.
Base of Logarithm
The base of a logarithm is a crucial part of understanding logarithmic functions. The base tells us which number is repeatedly multiplied to get a specific number. In the example of converting \( 10^{a^{2} b} = 97 \) into a logarithmic form, the base of the exponential function, which is 10, becomes the base of the logarithm.
This process uses the property that any exponential equation \( b^y = x \) can be converted to the logarithmic form \( y = \log_b{x} \).
  • **Base in exponential form**: In a term like \( 10^{a^{2}b} \), 10 is the base.
  • **Base in logarithm**: For \( \log_{10}{97} \), the base 10 indicates that we find out how many times you multiply 10 to reach 97.
Understanding the base helps us determine the repetitive factor in logarithmic and exponential functions, simplifying the interpretation of complex equations.
Properties of Logarithms
Logarithms are rich with properties that simplify exponential equations into something more manageable. The exercise showcases the conversion of an exponential equation to a logarithmic form using these properties. Key properties include:
  • **Change of Base**: Allows calculation of logarithms with bases other than 10 or e, by transforming it into a ratio of two logarithms.
  • **Product, Quotient, and Power Rules**: Logarithms turn products into sums, quotients into differences, and powers into multiplications, such as \( \log_b{(xy)} = \log_b{x} + \log_b{y} \).

In this exercise, we specifically used the idea that converting from exponential form to logarithmic form requires us to express the exponent as a logarithm. The property \( b^y = x \) becomes \( y = \log_b{x} \). This capability makes logarithms powerful, helping us deal with exponential growth or decay by turning multiplication and division problems into addition and subtraction ones.

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Most popular questions from this chapter

The dollar value of two investments after \(t\) years is given by \(f(t)=5000(1.062)^{t}\) and \(g(t)=\) \(9500(1.041)^{t} .\) Solve the equation \(f(t)=g(t) .\) What does your solution tell you about the investments?

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