Problem 4 - Contemporary Precalculus

Question

Find the logarithm, without using a calculator. $$\log \sqrt[3]{.01}$$

Step-by-Step Solution

Answer

Answer: The value of the expression $$\log \sqrt[3]{0.01}$$ is $$\frac{-2}{3}$$.

Step 1: Rewrite the cube root expression as an exponent

We can rewrite the cube root expression as an exponent: $$\sqrt[3]{0.01} = 0.01^{\frac{1}{3}}$$

Step 2: Apply the logarithm property

Now, apply the logarithm property $$\log{a^b} = b\log{a}$$ to the given expression: $$\log (0.01^{\frac{1}{3}}) = \frac{1}{3} \log (0.01)$$

Step 3: Express 0.01 as a power of 10

Since 0.01 is equal to \(10^{-2}\), we can rewrite the logarithm as follows: $$\frac{1}{3} \log (0.01) = \frac{1}{3} \log(10^{-2})$$

Step 4: Apply the logarithm property again

Now, apply the logarithm property $$\log{a^b} = b\log{a}$$ again: $$\frac{1}{3} \log(10^{-2}) = (-2) \cdot \frac{1}{3} \cdot \log{10}$$

Step 5: Evaluate the logarithm

Since the logarithm of 10 in base-10 is 1 (\(\log{10} = 1\)), we can simplify the expression: $$(-2) \cdot \frac{1}{3} \cdot \log{10} = (-2) \cdot \frac{1}{3} \cdot 1 = \frac{-2}{3}$$ So the final answer to the given exercise is: $$\log \sqrt[3]{0.01} = \frac{-2}{3}$$

Key Concepts

Exponents

Exponents are a great way to express repeated multiplication of a number by itself. They make it easier to work with very large or very small numbers. When you have a number like 10 raised to a power, the exponent tells you how many times the number is multiplied by itself. For example, \(10^3\) is \(10 \times 10 \times 10\), which equals 1000.
When working with numbers smaller than one, such as 0.01, exponents take on a negative value. So, 0.01 can be expressed as \(10^{-2}\), because it is like saying \(\frac{1}{10 \times 10}\). This is a powerful way to work with small decimals.

• Positive exponents show multiplication.
• Negative exponents show division by that many factors of ten.
• Using exponents can simplify calculations significantly.

Properties of Logarithms

Logarithms help us answer the question: to what power should a number, called the base, be raised to produce another number? This is particularly helpful when dealing with exponential growth or decay. A fundamental property of logarithms is \(\log(a^b) = b\log(a)\), which enables us to move exponents out front in a more manageable form.
This property is especially useful when dealing with complex equations involving roots or powers of non-standard bases, such as fractions or decimals, like in the example of \(\log(0.01^{1/3})\).

• You can simplify multiplication inside a logarithm by turning it into addition: \(\log(ab) = \log(a) + \log(b)\).
• You can simplify division inside a logarithm by turning it into subtraction: \(\log(\frac{a}{b}) = \log(a) - \log(b)\).
• A common base for logarithms in everyday use is 10, called the common logarithm.

Powers of Ten

Powers of ten are essential for simplifying the expressions of numbers, whether huge or minuscule. The system relies on using a base of 10 raised to various exponents but can drastically change the value expressed. For example, \(10^2\) is 100 and \(10^{-2}\) is 0.01.
In the realm of logarithms, recognizing powers of ten allows you to quickly evaluate expressions because the logarithm of 10 with base 10 is always 1. This is a key point: understanding that \(\log_{10}(10^x) = x\) helps to instantly evaluate many logarithmic expressions.

• Powers of ten are used extensively in science, such as in scientific notation.
• They provide a quick way to express the scale or size of a number.
• This concept ties directly into understanding real-world phenomena, from distances in space to microscopic measurements.