Problem 67 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 67-68: Approximate the function at the value of \(x\) to four decimal places. $$ h(x)=\log _{4} x-2 \log _{8} 1.2 x ; \quad x=5.3 $$

Step-by-Step Solution

Answer

The approximation of the function is approximately -0.5776.

Step 1: Express Logarithms with Common Base

First, we need to express all logarithms in the same base. Convert the logarithms to base 10 using the change of base formula: \[\log_{b} a = \frac{\log_{10} a}{\log_{10} b}\]. Thus, we have:\[\log_{4} x = \frac{\log x}{\log 4} \quad \text{and} \quad \log_{8} 1.2x = \frac{\log (1.2x)}{\log 8}\].

Step 2: Substitute the Values

Substitute \(x = 5.3\) into the function. We get:\[\log_{4} 5.3 - 2 \log_{8} (1.2 \times 5.3)\].

Step 3: Simplify Logarithmic Expressions

Using the results from Step 1, express each logarithm as:\[\log_{4} 5.3 = \frac{\log 5.3}{\log 4}\]and \[\log_{8} (1.2 \times 5.3) = \frac{\log (6.36)}{\log 8}\].Substituting these back, the function becomes:\[\frac{\log 5.3}{\log 4} - 2\left(\frac{\log 6.36}{\log 8}\right)\].

Step 4: Calculate the Logarithms

Use a calculator to find the logarithms:- For \(\log 5.3\) use: \(\log 5.3 \approx 0.7242\),- For \(\log 4\) use: \(\log 4 \approx 0.6021\),- For \(\log 6.36\) use: \(\log 6.36 \approx 0.8039\),- For \(\log 8\) use: \(\log 8 \approx 0.9031\).

Step 5: Substitute and Solve

Now substitute these values back into the equation:\[\frac{0.7242}{0.6021} - 2 \left( \frac{0.8039}{0.9031} \right)\].Calculate each part separately:- \(\frac{0.7242}{0.6021} \approx 1.203\),- \(\frac{0.8039}{0.9031} \approx 0.8903\),- Then, the expression is:\[1.203 - 2 \times 0.8903 = 1.203 - 1.7806\].

Key Concepts

Change of Base Formula

To understand logarithmic functions, it's essential to grasp the change of base formula. This formula allows us to convert any logarithm to a different base, which often simplifies calculations. The general formula is:\[ \log_{b} a = \frac{\log_{10} a}{\log_{10} b} \]
This formula states that you can express a logarithm with any base by using the common logarithms (base 10) or natural logarithms (base \(e\)).
By using base 10, you can use a standard scientific calculator to evaluate the logarithmic expressions easily. For example, in our exercise, we changed \( \log_{4}{5.3} \) to \( \frac{\log_{10} 5.3}{\log_{10} 4} \) which simplifies the computation. Breaking down complex logarithms into simpler base 10 format enables more straightforward calculation without advanced tools, making it widely used in various mathematical contexts.

Logarithmic Approximation

Logarithmic approximation is incredibly useful in practical computations where exact values might be challenging to calculate manually. This is especially true when dealing with non-integer values of \(x\) in logarithmic functions.
When you approximate a logarithm, you round it to a certain number of decimal places depending on the precision required. In our exercise, the task was to approximate each component of the function to four decimal places using a calculator:

• \(\log 5.3 \approx 0.7242\)
• \(\log 4 \approx 0.6021\)
• \(\log 6.36 \approx 0.8039\)
• \(\log 8 \approx 0.9031\)

After finding these approximations, we are able to compute the original logarithmic expression accurately. This method helps when calculators only provide limited precision, ensuring your results remain reliable during step-by-step computational tasks.

Common Logarithms

Common logarithms are logarithms with base 10, denoted simply by \(\log\). They are one of the most widely used types of logarithms due to the ease of computation, as most calculators are equipped to handle \(\log\) operations easily.
This versatility allows us to solve a broad range of mathematical problems, including those in algebra and calculus, that involve logarithms. For instance, in our exercise, all logarithmic terms were converted to common logarithms to simplify calculations with the change of base formula.
Common logarithms not only make calculations swifter but also enable us to work without worrying about different logarithmic bases. They are integral to our ability to perform several mathematical tasks, ensuring that we can always find a solution where logarithms are present.