Problem 32 $g(x)=\log _{6} x$... [FREE SOLUTION]

91影视

Trigonometry

Ron Larson, Robert Hostetler

$Math Studyset 91影视 Explanations$ Math

7 Edition

Chapter 5: Problem 32

$g(x)=\log _{6} x$

Short Answer

Expert verified

The value of x in the function $g(x) = \log _{6} x$ is equivalent to the value you'd get if you raise 6 to the power of n (where n is a real number). So, the function can be expressed as an exponential function, $6^n = x$.

Step by step solution

Understand the Function

Before we can solve the function, we need to understand what $\log _{6} x$ means. When we see $\log _{6} x$ , it's asking the question '6 to the power of what equals x?' or more mathematically 'what power do we raise 6 to in order to get x?' Thus, x is a positive number greater than zero.

Solve the Function

Now, let's suppose we have the equation $\log _{6} x = n$. This equation represents 'what power do we raise 6 to in order to get x?' It may be easier to solve this equation if we convert the log expression into exponential form. The exponential form of the equation is $6^n = x$. So, which ever value our 'n' in the equation $\log _{6} x = n$ is, that same value would be our 'x' in the equation $6^n = x$.

Understand the Result

It's important to understand that the result x is not a single value, but rather will change dependent on the value of n (the exponent we raise 6 to). This is because the function g(x) presents a mathematical rule or relationship, rather than a solid algebraic equation. As a result, the 'solution' for $g(x)$ is actually an expression which allows us to calculate x from a given exponent n (provided that it is a real number).

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Key Concepts

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

Understanding Logarithms

Logarithms, often just referred to as 'logs', are the inverse operations to exponentiation. They answer the question 'to what power must a base be raised to produce a certain number?'. In the equation g(x) = $log_6 x$, g(x) gives us the exponent that 6 must be raised to in order to get x.

For example, if we have $log_6 36 = 2$, this means that raising 6 to the power of 2 results in 36. Logarithms have a few important rules, such as the product, quotient, and power rules, which can make working with them easier. Recognizing these patterns is essential in solving logarithmic equations efficiently.

Logarithmic Equations

Solving logarithmic equations involves finding the value of the variable that makes the equation true. It often requires us to use properties of logarithms and to transform the log equation into its equivalent exponential form. When we're faced with an equation like $log_6 x = n$, we're essentially being asked to figure out to what power we must raise 6 to get x. If we know the value of n, switching to exponential form gives us a straightforward answer.

In dealing with logarithmic equations, it is critical to remember that the base of the log and the resultant value must be positive numbers, as logs are not defined for zero or negative bases and results.

Exponential Form Conversion

When we convert a logarithmic equation to its exponential form, we're using the simple principle that if $log_b a = c$, then $b^c = a$. This shift not only makes the relationship between the variables clear but also often simplifies the process of finding the value of unknowns. For our function g(x), the exponential form $6^n = x$ shows us directly how x will change based on different values of n.

Understanding how to switch between logarithmic and exponential forms is crucial for solving problems related to exponential growth or decay, computational complexity, and in financial calculations involving compound interest.

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91影视

\(g(x)=\log _{6} x\)

Short Answer

Step by step solution

Understand the Function

Solve the Function

Understand the Result

Key Concepts

Understanding Logarithms

Logarithmic Equations

Exponential Form Conversion

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