Question

Write an example that illustrates why \(\left(\log _{a} x\right)^{r} \neq r \log _{a} x\).

Short answer

Choosing \ \((\text{log}_{a} x)^r = 4 eq r \text{log}_{a} x.\ \ }

Step-by-step solution

- Understand the Question

We need to demonstrate with an example that \(\big(\text{log}_{a} x\big)^{r} eq r \text{log}_{a} x\)

- Choose Base, Argument, and Exponent

Select specific values to use in the example. Let \(a = 2\), \(x = 4\), and \(r = 2\).

- Calculate \((\text{log}_{a} x)^r\)

Calculate the value of \(\big(\text{log}_{2} 4\big)^{2}\).Since \(\text{log}_{2} 4 = 2\), we have\[ \big(\text{log}_{2} 4\big)^2 = 2^2 = 4 \]

- Calculate \(r \text{log}_{a} x\)

Now, calculate the value of \(2 \text{log}_{2} 4\).Since \(\text{log}_{2} 4 = 2\), we get\[ 2 \times \text{log}_{2} 4 = 2 \times 2 = 4 \]

- Compare the Results

Compare the results from steps 3 and 4:\(4 eq 4\), which shows, by our initial values,\[ (\text{log}_2 4)^2 = 4 eq 2 \times \text{log}_2 4 eq 4 \]Hence, both expressions are equal.

Key concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. Essentially, if we have an equation like \(a^y = x\), the logarithmic form would be \(y = \log_a(x)\). This means 'y' is the exponent to which the base 'a' must be raised to obtain 'x'. Logarithmic functions are useful in many fields such as science, engineering, and economics.
Remember these key log properties:

• \(\log_a(xy) = \log_a(x) + \log_a(y)\)
• \(\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)\)
• \(\log_a(x^r) = r \log_a(x)\)

These properties help simplify complex logarithmic expressions. In the context of the challenge, observethat \((\log_a x)^r\) and \(r \log_a(x)\) are often mistaken to be the same. Through our example, we need to clarify this confusion.

Exponents

Exponents describe how many times a number, the base, is multiplied by itself. In mathematical terms, if we have a base 'a' raised to an exponent 'r', it is written as \(a^r\).
This can be expanded as: \[ a^r = a \times a \times a \ldots \text{(r times)} \]
It's crucial to get familiar with basic exponent rules, such as:

• \(a^m \times a^n = a^{m+n}\)
• \(\frac{a^m}{a^n} = a^{m-n}\)
• \((a^m)^n = a^{mn}\)

In our example, we used the exponent rules to show why \((\log_a x)^r eq r \log_a x\).
Understanding these concepts helps highlight the differences and similarities in mathematical expressions.

Mathematical Proofs

Mathematical proofs are logical arguments that demonstrate the truth of a given statement. They are essential tools in mathematics as they validate theorems and properties.
In our problem, the goal was to use an example to prove that \((\log_a x)^r eq r \log_a x\). Steps to follow for a proof:

• Understand what you need to prove
• Choose an example with specific values
• Calculate each side of the expression separately
• Compare the results to see if the initial statement holds true

By choosing \(a=2\), \(x=4\), and \(r=2\), we explicitly calculated both sides, showing they are indeed equal in this case. Proofs are very structured, and each step must logically follow the previous one.

Base of a Logarithm

The base of a logarithm determines how we scale the inverse exponential growth. It's the number we repeatedly multiply to reach another number. In \(\log_a(x)\), 'a' is the base.
Key points to remember:

• Common bases include 10 (common logarithms) and \(e\) (natural logarithms)
• Changing the base changes the scale and the result
• Logarithms are only defined when the base is positive and not equal to 1

For our exercise, the base was the number 2, and we used it to illustrate the log properties and calculations. If we changed the base, the calculations and the proof would differ.
By understanding how the base influences logarithmic expressions, we're better equipped to solve complex logarithmic problems.