Question

The graph of \(f(x)=\log _{8} x\) can also be described by the equation \(g(x)=a \log _{2} x\) Find the value of \(a\).

Short answer

The value of \(\ta\) is \(\t\frac{1}{3}\).

Step-by-step solution

- Understand the Change of Base Formula

The change of base formula for logarithms states that \ \[ \log_b x = \frac{\log_k x}{\log_k b} \ \] where 'b' and 'k' are the bases of the logarithm and the new base respectively.

- Apply the Change of Base Formula to \(\tlog_{8} x\)

To convert \(\tlog_{8} x\) to base 2, use the change of base formula: \[ \log_{8} x = \frac{\log_{2} x}{\log_{2} 8} \ \]

- Simplify \(\tlog_{2} 8\)

We know that \(\t8 = 2^3\), so: \[ \log_{2} 8 = \log_{2} (2^3) = 3 \ \]

- Substitute and Simplify

Substitute \(\tlog_{2} 8\) in the change of base formula: \[ \log_{8} x = \frac{\log_{2} x}{3} \ \] This implies that \(\t\log_{8} x\) can be rewritten as \(\tg(x) = \frac{1}{3} \log_{2} x\).

- Identify the Value of \(\ta\)

Given \(\tg(x) = a \log_{2} x \), compare this with \(\tg(x) = \frac{1}{3} \log_{2} x\). This tells us that the value of \(\ta\) is \(\t\frac{1}{3}\).

Key concepts

Logarithmic Functions

Logarithms are a fundamental part of mathematics, especially in precalculus and beyond. Understanding logarithms is key because they help us solve exponential equations. A logarithmic function is the inverse of an exponential function. For example, if you have the function \(f(x) = b^x\), then the logarithmic function is \(g(x) = \log_b x\). This means if you know the exponent, you can find the number using the logarithm. Logarithmic functions have properties that make them useful for solving equations:

• \( \log_b (xy) = \log_b x + \log_b y \) - This is the product rule.
• \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \) - This is the quotient rule.
• \( \log_b (x^k) = k \log_b x \) - This is the power rule.

Understanding these rules is important for manipulating and solving logarithmic equations.

Base Conversion

Sometimes, it becomes necessary to change the base of a logarithmic function to make calculations easier. This is where the change of base formula comes into play:
\[\backslash log_b x = \frac{\log_k x}{\log_k b}\]
Here, \'b\' is the original base, while \'k\' is the new base. For example, in the given exercise, we needed to change the base of \log_{8} x\ to base 2. We use the change of base formula with \'x\' as the number we are taking the logarithm of, \'b\' as 8, and our new base \'k\' as 2:
\[\backslash log_{8} x = \frac{\log_{2} x}{\log_{2} 8}\]
Since 8 can be written as \2^3\, we know that \log_{2} 8 = 3\. Substitute this back into the formula:
\[\backslash log_{8} x = \frac{\log_{2} x}{3}\]
So the logarithm of x base 8 can be converted to a base 2 logarithm by dividing by 3.

Precalculus

Precalculus is the mathematical foundation you need to tackle higher-level calculus. It combines principles from algebra and trigonometry to prepare you for the complex concepts in calculus. Precalculus covers various functions, including exponential and logarithmic functions.
Understanding these subjects prepares you for more advanced topics in mathematics. For instance, in the step-by-step solution provided, we applied rules and formulas learned in precalculus:

• We used the definition of logarithms to apply the change of base formula.
• Recognized the need to express 8 as a power of 2.
• Finally, simplified the function to find our constant.

All these steps utilize skills honed in precalculus, solidifying your understanding and making you ready for calculus.