Question

Prove each identity. a) \(\log _{q^{3}} p^{3}=\log _{q} p\) b) \(\frac{1}{\log _{p} 2}-\frac{1}{\log _{q} 2}=\log _{2} \frac{p}{q}\) c) \(\frac{1}{\log _{q} p}+\frac{1}{\log _{q} p}=\frac{1}{\log _{q^{2}} p}\) d) \(\log _{\frac{1}{q}} p=\log _{q} \frac{1}{p}\)

Short answer

a) \(\log_{q^3} p^3 = \log_q p\); b) \(\frac{1}{\log_p 2} - \frac{1}{\log_q 2} = \log_2 \frac{p}{q}\); c) \(\frac{2}{\log_q p} = \frac{1}{\log_{q^2} p}\); d) \(\log_{\frac{1}{q}} p = \log_q \frac{1}{p}\).\)

Step-by-step solution

Understand the given logarithmic identity (a)

The identity to prove is \(\log_{q^3} p^3 = \log_q p\).

Apply the change of base formula

Rewrite \(\log_{q^3} p^3\) using the change of base formula: \[\log_{q^3} p^3 = \frac{\log_q p^3}{\log_q q^3}\.\]

Simplify the equation

Simplify each logarithm separately: \[\log_q p^3 = 3\log_q p\] and \[\log_q q^3 = 3\log_q q = 3\.\] Thus, \[\frac{3\log_q p}{3} = \log_q p\.\]

Conclusion for part (a)

Therefore, \(\log_{q^3} p^3 = \log_q p\).

Understand the given logarithmic identity (b)

The identity to prove is \[\frac{1}{\log_p 2} - \frac{1}{\log_q 2} = \log_2 \frac{p}{q}\.\]

Use the change of base formula

Using the change of base formula, we know \[\frac{1}{\log_a b} = \log_b a\.\] This converts our logs to: \[\log_2 p - \log_2 q\.\]

Apply the property of logarithms

Use the property \[\log_a b - \log_a c = \log_a \frac{b}{c}\] to get: \[\log_2 p - \log_2 q = \log_2 \frac{p}{q}\.\]

Conclusion for part (b)

Therefore, \[\frac{1}{\log_p 2} - \frac{1}{\log_q 2} = \log_2 \frac{p}{q}\.\]

Understand the given logarithmic identity (c)

The identity to prove is \[\frac{1}{\log_q p} + \frac{1}{\log_q p} = \frac{1}{\log_{q^2} p}\.\]

Simplify the left-hand side

Combine the fractions: \[\frac{1}{\log_q p} + \frac{1}{\log_q p} = 2\frac{1}{\log_q p} \Rightarrow \frac{2}{\log_q p}\.\]

Rewrite the right-hand side with the change of base formula

Using the change of base formula: \[\log_{q^2} p = \frac{\log_q p}{\log_q q^2} = \frac{\log_q p}{2\log_q q} = \frac{\log_q p}{2}\.\]

Conclusion for part (c)

Therefore, \[\frac{2}{\log_q p} = \frac{1}{\log_{q^2} p}\.\]

Understand the given logarithmic identity (d)

The identity to prove is \[\log_{\frac{1}{q}} p = \log_q \frac{1}{p}\.\]

Apply the change of base formula

Use the change of base formula: \[\log_{\frac{1}{q}} p = \frac{\log p}{\log \frac{1}{q}}\.\]

Simplify the logarithm

Use the property \[\log \frac{1}{b} = -\log b\] to get: \[\log_{\frac{1}{q}} p = \frac{\log p}{-\log q} = -\frac{\log p}{\log q} = \log_q \frac{1}{p}\.\]

Conclusion for part (d)

Therefore, \[\log_{\frac{1}{q}} p = -\log_q p = \log_q \frac{1}{p}\.\]

Key concepts

change of base formula

The change of base formula is a very important tool in logarithms. It allows us to convert a logarithm with any base into a logarithm with a different base. The formula states: \ \[ \log_b a = \frac{\log_c a}{\log_c b} \]. This means that you can express \log_b a using any new base 'c'. You often use this formula to simplify logarithmic expressions or to switch to a base that is easier to handle. For example, in part (a) of the exercise, the expression \log_{q^3} p^3 was converted to \[ \log_{q^3} p^3 = \frac{\log_q p^3}{\log_q q^3} \], allowing us to simplify and prove the identity.Using this formula can make many logarithmic problems simpler.

properties of logarithms

Logarithms have several properties that make working with them easier. One key property is the power rule: \[ \log_b (a^c) = c \log_b a \]. For example, \log_q p^3 can be written as 3 \log_q p. Another useful property is the product and quotient rules: \[ \log_b (xy) = \log_b x + \log_b y \] and \[ \log_b \frac{x}{y} = \log_b x - \log_b y \]. In part (b) of the exercise, we use the change of base formula and then apply these properties to combine terms: \[ \log_2 \frac{p}{q} = \log_2 p - \log_2 q \]. Understanding these properties makes it easier to manipulate and simplify logarithmic expressions.

logarithmic simplification

Simplifying logarithmic expressions often involves changing the form of the logarithm using the properties of logarithms. For example, in part (c), we used properties to combine fractions: \[ \frac{1}{\log_q p} + \frac{1}{\log_q p} = 2 \cdot \frac{1}{\log_q p} = \frac{2}{\log_q p} \]. This was then compared to another simplified expression obtained using the change of base formula. Simplification can help prove identities and solve equations by making the expressions more manageable.It’s important to practice these steps to become proficient at logarithmic simplification.

logarithmic equations

Solving logarithmic equations often involves using logarithmic properties and simplification techniques. For example, in part (d), we start by using the change of base formula and then simplify: \[ \log_{\frac{1}{q}} p = \frac{\log p}{\log \frac{1}{q}} \]. By applying the property \[ \log \frac{1}{b} = -\log b \], we then get \[ -\frac{\log p}{\log q} \], which simplifies to \log_q \frac{1}{p}. By transforming the equation step-by-step and using logarithmic properties, we arrive at the solution. Being methodical and understanding the properties is key to solving logarithmic equations effectively.