Question

If \(\log _{8} M=\frac{\log _{5} 7}{\log _{5} 8},\) then \(M=\) ______________

Short answer

7

Step-by-step solution

Understand the Given Equation

The problem provides \(\text{log}_{8} M = \frac{\text{log}_{5} 7}{\text{log}_{5} 8}\). This equation involves logarithms with different bases, which suggests that a change of base formula might be useful.

Apply the Change of Base Formula

Recall the change of base formula: \(\text{log}_{b} a = \frac{\text{log}_{c} a}{\text{log}_{c} b}\). Apply it to the right-hand side to express it in terms of logarithms with base 8. \(\frac{\text{log}_{5} 7}{\text{log}_{5} 8} = \text{log}_{8} 7\).

Equate Logarithms

Now we have \(\text{log}_{8} M = \text{log}_{8} 7\). Since the logarithms are equal, their arguments must be equal. Therefore, \(M = 7\).

Key concepts

change of base formula

When working with logarithms, you often need to convert between different bases. This is where the change of base formula comes in handy. The formula states that for any positive numbers \(a\), \(b\), and \(c\), where \(a\) and \(b\) are not equal to 1, you can express \( \text{log}_{b} a \) using a different base: \[ \text{log}_{b} a = \frac{\text{log}_{c} a}{\text{log}_{c} b} \] This means you can choose any base \(c\) common to the logarithms in the numerator and denominator.

In many textbooks and problems, the base \(10\) or base \(e\) (natural logarithm) is used because calculators provide these functions. In our exercise, we had a logarithmic expression using base 5, and we needed to compare it with base 8. Here, we applied the change of base formula to convert the given expression into one easily comparable with base 8.

logarithmic equations

Logarithmic equations are equations that involve logarithms with the variable inside the logarithm. Solving these requires a good understanding of the logarithm properties and rules.

Consider the equation given in the exercise: \(\text{log}_{8} M = \frac{\text{log}_{5} 7}{\text{log}_{5} 8}\). This is a classic logarithmic equation where the log expressions on both sides ultimately need to be simplified and compared.

The key to solving these equations lies in simplifying the expression by using formulas and properties like the change of base formula, product rule, quotient rule, and power rule of logarithms as needed. Once simplified, logarithmic equality tells us that if two logs with the same base are equal, their arguments must also be equal.

solving logarithms

Solving logarithmic equations involves a few clear steps. Consider the given example to illustrate the process: If \( \text{log}_{8} M=\frac{\text{log}_{5} 7}{\text{log}_{5} 8} \), find \( M \).

• Step 1: Simplify using the change of base formula
  First, recognize that the right-hand side can be simplified using the change of base formula: \[ \frac{\text{log}_{5} 7}{\text{log}_{5} 8} = \text{log}_{8} 7 \] This reduces the equation to \(\text{log}_{8} M = \text{log}_{8} 7\).
• Step 2: Equate the arguments
  Simplifying further, we use the property that if \( \text{log}_{b} a = \text{log}_{b} c \), then \( a = c \). Thus, \( M = 7 \).

By following these logical steps and applying logarithmic properties, you can solve similar log equations efficiently. Remember to check for extraneous solutions in more complex scenarios.