Question

Assume \(\log _{b} x=0.36, \log _{b} y=0.56\), and \(\log _{b} z=0.83 .\) Evaluate the following expressions. $$\log _{b} \frac{\sqrt{x}}{\sqrt[3]{z}}$$

Short answer

Answer: The value of the expression \(\log_b \frac{\sqrt{x}}{\sqrt[3]{z}}\) is approximately \(-0.0966\).

Step-by-step solution

Rewrite the expression using logarithm properties

Using the properties of logarithms mentioned in the analysis, we can simplify the given expression: $$\log_b \frac{\sqrt{x}}{\sqrt[3]{z}} = \log_b \left(\frac{x^{\frac{1}{2}}}{z^{\frac{1}{3}}}\right)$$ Applying the third logarithm property, we get: $$\log_b \left(\frac{x^{\frac{1}{2}}}{z^{\frac{1}{3}}}\right) = \log_b x^{\frac{1}{2}} - \log_b z^{\frac{1}{3}}$$ Now, applying the first logarithm property to both terms, we have: $$\frac{1}{2}\log_b x - \frac{1}{3}\log_b z$$

Substitute the given values

Now, we substitute the given values of \(\log_b x\) and \(\log_b z\) in the expression: $$\frac{1}{2}(\log_b x) - \frac{1}{3}(\log_b z) = \frac{1}{2}(0.36) - \frac{1}{3}(0.83)$$

Calculate the value of the expression

Finally, we calculate the value of the expression: $$\frac{1}{2}(0.36) - \frac{1}{3}(0.83) = \frac{0.36}{2} - \frac{0.83}{3} = 0.18 - 0.2766 \approx -0.0966$$ So, the value of the expression \(\log_b \frac{\sqrt{x}}{\sqrt[3]{z}} \approx -0.0966\).

Key concepts

Properties of Logarithms

Logarithms have several properties that make them very helpful in simplifying expressions. One of the main properties is the **product property**. When you multiply two numbers, their logarithm can be expressed as the sum of the logarithms of the numbers:

• \( \log_b(xy) = \log_b x + \log_b y \).

Another essential property is the **quotient property**. Similar to multiplication, division can be handled by taking the difference between the logarithms:

• \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \).

The last major property to understand is the **power property**. If you have an exponent on a number, you can "bring it down" in front of the logarithm:

• \( \log_b(x^n) = n \cdot \log_b x \).

These properties are really handy. They turn multiplicative and divisive situations into more straightforward additions and subtractions. With these, complex expressions become manageable.
Try to practice these properties to gain confidence in handling logarithms by turning complex operations grounded upon them into simpler operations.

Evaluating Logarithmic Expressions

Being able to evaluate logarithmic expressions is crucial when working with logs. To evaluate anything with logs, you should first try to simplify using the properties of logarithms.
The exercise you just saw had a complicated expression \( \log_b \left( \frac{\sqrt{x}}{\sqrt[3]{z}} \right) \). By using the quotient and power properties (as shown in step 1), this turned into a much simpler form.

• First, rewrite roots as exponents: \( \sqrt{x} = x^{1/2} \) and \( \sqrt[3]{z} = z^{1/3} \).
• Then apply the power property: \( \log_b(x^{1/2}) = \frac{1}{2} \log_b x \) and \( \log_b(z^{1/3}) = \frac{1}{3} \log_b z \).
• Apply the quotient property: \( \log_b \left( \frac{x^{1/2}}{z^{1/3}} \right) = \frac{1}{2} \log_b x - \frac{1}{3} \log_b z \).

Now, plug in the known values of \( \log_b x \) and \( \log_b z \).
This approach illustrates how step-by-step application of properties allows for more effortless evaluations.

Logarithmic Identities

In addition to the main properties, logarithmic identities serve as shortcuts and additional tools when simplifying or calculating. One identity is that the logarithm of 1 to any base is always zero:

• \( \log_b 1 = 0 \)

This is because any base raised to the power of zero gives 1.
Another important identity is that when the logarithm and exponent share the same base, they cancel each other out:

• \( b^{\log_b x} = x \)
• \( \log_b b^x = x \)

These identities are particularly useful for simplifying expressions quickly.
Knowing these makes it easier to calculate expressions without always relying on a calculator, fostering a deeper understanding of the interactions between logarithms and exponents. Each identity builds your skills in recognizing familiar patterns and finding shortcuts in mathematical reasoning and problem-solving.