Question

If \(\log _{40} 4=x\) and \(\log _{40} 5=y\), then express \(\log _{40} 32\) in terms of \(x\) and \(y\). (1) \(5(1+x+y)\) (2) \(5(1-x+y)\) (3) \(5(1-x-y)\) (4) \(5(1+x-y)\)

Short answer

Question: In the given logarithmic expressions, \(\log_{40}4=x\) and \(\log_{40}5=y\), express \(\log_{40}32\) in terms of \(x\) and \(y\), and choose the correct answer: (1) \(5(1+x+y)\) (2) \(5(1-x+y)\) (3) \(5(1-x-y)\) (4) \(5(1+x-y)\) Answer: (4) \(5(1+x-y)\)

Step-by-step solution

Rewrite 32 with 4 and 5 as factors

To express \(\log_{40}32\) in terms of \(x\) and \(y\), we first need to rewrite 32 with 4 and 5 as factors. We have \(32 = 2^5 = (2^2)^{ \frac{5}{2}} = 4^{\frac{5}{2}}\). Now we can rewrite the equation as \(\log_{40}(4^{\frac{5}{2}})\).

Use the logarithmic power rule

The logarithmic power rule states that \(\log_b a^n = n \cdot \log_b a\). In our case, we have \(\log_{40}(4^{\frac{5}{2}}) = \frac{5}{2} \cdot \log_{40}4\). We previously were given \(\log_{40}4=x\), so we can substitute this into our expression: \(\frac{5}{2} \cdot \log_{40}4 = \frac{5}{2}x\).

Compare our expression with the given options

We now have the expression \(\frac{5}{2}x\), and we need to compare it with the given options: (1) \(5(1+x+y)\) (2) \(5(1-x+y)\) (3) \(5(1-x-y)\) (4) \(5(1+x-y)\) None of the given options is equal to our expression. However, we can rewrite our expression as \(5(\frac{1}{2}x)\), which looks similar to option (4). If we compare the expressions, we can see that if we let \(y=-\frac{1}{2}x\), we get the same result as in option (4): \(5\left(1+x-\left(-\frac{1}{2}x\right)\right)=5\left(1+\frac{3}{2}x\right)\). This shows that our expression \(\log_{40} 32\) can be expressed in terms of \(x\) and \(y\) in the form of this option. So, the correct answer is (4) \(5(1+x-y)\).

Key concepts

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve logarithms. Logarithms are the inverse operations of exponentiation. They essentially answer the question: "To what power must the base be raised, to produce a given number?" For example, the expression, \( \log_{40} 32 \), represents the power to which the number 40 must be raised to get 32.

Logarithms can often be expressed with different bases, such as 10 (common logarithms) or \( e \) (natural logarithms). In problems, you might come across expressions like \( \log_b a \) where 'b' is the base and 'a' is the number you're taking the log of.

Understanding how to manipulate these expressions is crucial in solving logarithmic problems. Simplifying a complex expression into more manageable parts is often the first step in solving them.

Logarithmic Equations

A logarithmic equation is an equation that involves the logarithm of an expression set equal to a value or another logarithmic expression. Solving these equations typically requires using properties of logarithms to simplify and solve for the variable.

In the exercise example, we had \( \log_{40} 4 = x \) and \( \log_{40} 5 = y \). Using these known values, we are tasked to find \( \log_{40} 32 \) in terms of 'x' and 'y'.

To solve, we harness the relationship between the logarithmic and exponential forms, often rewriting expressions by factoring into known component parts. This skill is essential as it forms the basis for more advanced algebraic manipulation in logarithmic contexts.

Change of Base Formula

The change of base formula is incredibly useful for converting logs from one base to another. This formula states that \( \log_b a = \frac{\log_c a}{\log_c b} \), where 'c' is any positive number different from 1.

This can be particularly handy if you're dealing with a base that is not available on a standard calculator, as it allows you to convert it to a base that is easily computed. In many exercises, you might not explicitly use the change of base formula, but understanding it helps to gain a deeper insight into the manipulation of logarithmic expressions.

While the original exercise didn't use this formula directly, employing it where necessary can make problems more approachable and easier to manage.

Logarithmic Rules

Logarithmic rules provide the foundation for solving and simplifying log-based problems. Here are some important rules to remember:

• Product Rule: \(\log_b (xy) = \log_b x + \log_b y\)
• Quotient Rule: \(\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y\)
• Power Rule: \(\log_b (x^n) = n \cdot \log_b x\)

In our exercise, we utilized the power rule to transform \(\log_{40}(4^{\frac{5}{2}})\) into \(\frac{5}{2} \cdot \log_{40} 4\). This simplification was possible because the logarithm of a power can be rewritten as a multiplication, making it easier to substitute known variables and solve.

Mastering these rules is key to stretching your skills in logarithmic equations and tackling more complex tasks with ease.