Problem 83
Question
Evaluate or simplify each expression without using a calculator. $$ \log 10^{7} $$
Step-by-Step Solution
Verified Answer
The value of the expression \(\log 10^{7}\) is 7.
1Step 1: Understanding Logarithms
The logarithm is the inverse of the exponentiation. It means that if we have \(\log_a b = c\), then this can be re-written as \(a^c = b\). This property of logarithms is what allows us to solve this exercise.
2Step 2: Applying the definition to our problem
In this problem, our base \(a\) is 10 (the base of common logarithms), \(b\) is \(10^7\), and we are asked to solve for \(c\), the output of the log function. Using the definition from step 1, we find that \(10^c = 10^7\).
3Step 3: Solving for the unknown
In order for \(10^c\) to be equal to \(10^7\), \(c\) has to be equal to 7. Which means that \(\log 10^{7} = 7\).
Key Concepts
Logarithmic FunctionsExponentiationSolving Logarithmic Equations
Logarithmic Functions
Logarithmic functions are a crucial part of understanding how various phenomena operate on a non-linear scale. At their core, these functions answer the question: 'To what power must we raise a base number to obtain another number?'. Mathematically, this is expressed as \( \log_b(x) = y \) if and only if \( b^y = x \). Here, \( b \) is the base of the logarithm, \( x \) is the number we are taking the logarithm of, and \( y \) is the exponent to which the base is raised to result in \( x \).
For instance, in the expression \( \log(10^7) \) from the exercise, this reads as 'to what power must 10 be raised to yield \( 10^7 \)?' Since the answer is intuitively 7, the value of the logarithm is 7. The efficiency of logarithmic functions becomes more apparent when dealing with large numbers or complex computations, demonstrating their transformative nature in simplifying expressions and calculations.
For instance, in the expression \( \log(10^7) \) from the exercise, this reads as 'to what power must 10 be raised to yield \( 10^7 \)?' Since the answer is intuitively 7, the value of the logarithm is 7. The efficiency of logarithmic functions becomes more apparent when dealing with large numbers or complex computations, demonstrating their transformative nature in simplifying expressions and calculations.
Exponentiation
Exponentiation is the mathematical operation involving two numbers, where one number (the base) is raised to the power of the other (the exponent). If we have \( b \) as a base and \( n \) as an exponent, the exponentiation is written as \( b^n \). It signifies the base \( b \) multiplied by itself \( n \) times.
Understanding exponentiation is vital since it's the process that logarithms reverse. When we solve exponential equations, we often find the logarithm to be a handy tool for finding the unknown exponent. It's fascinating to see the interplay between the two operations. In the given exercise, the exponentiation comes into play by recognizing that the base 10 is raised to the power of 7, making the task of finding the logarithm straightforward.
Understanding exponentiation is vital since it's the process that logarithms reverse. When we solve exponential equations, we often find the logarithm to be a handy tool for finding the unknown exponent. It's fascinating to see the interplay between the two operations. In the given exercise, the exponentiation comes into play by recognizing that the base 10 is raised to the power of 7, making the task of finding the logarithm straightforward.
Solving Logarithmic Equations
Solving logarithmic equations can initially seem daunting, but by grasping the relationship between logarithms and exponentiation, it becomes much more manageable. To solve a logarithmic equation, we often look for an exponent that satisfies the equation as portrayed by the logarithm. This involves understanding the concept that \( \log_b(x) = y \) translates into \( b^y = x \).
When faced with an equation such as \( \log(10^7) \), the strategy is to match the form of the number inside the logarithm with the base, making it evident that the exponent is our solution. In this case, identifying that \( 10^7 = 10^7 \) is relatively simple, leading to the conclusion that \( c = 7 \). This illustrates the beauty of logarithms – they demystify the process of handling exponents by providing a method to 'undo' the exponentiation, making it easier to isolate and solve for variables.
When faced with an equation such as \( \log(10^7) \), the strategy is to match the form of the number inside the logarithm with the base, making it evident that the exponent is our solution. In this case, identifying that \( 10^7 = 10^7 \) is relatively simple, leading to the conclusion that \( c = 7 \). This illustrates the beauty of logarithms – they demystify the process of handling exponents by providing a method to 'undo' the exponentiation, making it easier to isolate and solve for variables.
Other exercises in this chapter
Problem 82
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