Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.

logb (√x) / (³√z)

Verified step by step guidance

Start by using the properties of logarithms to simplify the expression $ \log_b \left( \frac{\sqrt{x}}{\sqrt[3]{z}} \right) $.

Apply the quotient rule of logarithms: $ \log_b \left( \frac{\sqrt{x}}{\sqrt[3]{z}} \right) = \log_b (\sqrt{x}) - \log_b (\sqrt[3]{z}) $.

Use the power rule of logarithms: $ \log_b (\sqrt{x}) = \log_b (x^{1/2}) = \frac{1}{2} \log_b x $.

Similarly, apply the power rule to $ \log_b (\sqrt[3]{z}) = \log_b (z^{1/3}) = \frac{1}{3} \log_b z $.

Substitute the given values: $ \frac{1}{2} \log_b x = \frac{1}{2} \times 0.36 $ and $ \frac{1}{3} \log_b z = \frac{1}{3} \times 0.83 $, then subtract the two results.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Logarithms

The properties of logarithms include rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (logb(mn) = logb(m) + logb(n)), the quotient rule (logb(m/n) = logb(m) - logb(n)), and the power rule (logb(m^k) = k * logb(m)). Understanding these properties is essential for evaluating complex logarithmic expressions.

Change of Base Formula

The change of base formula allows the conversion of logarithms from one base to another, expressed as logb(a) = logk(a) / logk(b) for any positive k. This is particularly useful when dealing with logarithms of different bases, enabling easier calculations and comparisons. It is important for evaluating logarithmic expressions when the base is not easily manageable.

Radicals and Exponents

Radicals, such as square roots and cube roots, can be expressed in terms of exponents, where √x = x^(1/2) and ³√z = z^(1/3). This relationship is crucial when simplifying logarithmic expressions involving roots, as it allows the application of the power rule of logarithms. Recognizing this connection helps in transforming and evaluating logarithmic expressions effectively.

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