Question

fill in the missing values to make the equations true.\\(\log_{7}3 - \log_{7}4 = \log_{7}\square\\)\\(\log_{2}\square + \log_{2}3 = \log_{2}15\\)\\(\log_{7}81 = 4\log_{7}\square\\)

Explanation:

First Equation: $\boldsymbol{\log_{7} 3 - \log_{7} 4 = \log_{7} \square}$

Step1: Recall log subtraction rule

The logarithmic property for subtraction is $\log_b a - \log_b c = \log_b \frac{a}{c}$.

Step2: Apply the rule

Here, $a = 3$, $c = 4$, and $b = 7$. So $\log_{7} 3 - \log_{7} 4 = \log_{7} \frac{3}{4}$.

Second Equation: $\boldsymbol{\log_{2} \square + \log_{2} 3 = \log_{2} 15}$

Step1: Recall log addition rule

The logarithmic property for addition is $\log_b a + \log_b c = \log_b (a \times c)$.

Step2: Let the missing value be $x$

We have $\log_{2} x + \log_{2} 3 = \log_{2} 15$. Using the addition rule, this becomes $\log_{2} (3x) = \log_{2} 15$.

Step3: Solve for $x$

Since the logarithms have the same base and are equal, their arguments must be equal. So $3x = 15$. Dividing both sides by 3, we get $x = 5$.

Third Equation: $\boldsymbol{\log_{7} 81 = 4\log_{7} \square}$

Step1: Recall log power rule

The logarithmic property for exponents is $n\log_b a = \log_b (a^n)$.

Step2: Let the missing value be $x$

We have $4\log_{7} x = \log_{7} 81$. Using the power rule, this becomes $\log_{7} (x^4) = \log_{7} 81$.

Step3: Solve for $x$

Since the logarithms have the same base and are equal, their arguments must be equal. So $x^4 = 81$. Taking the fourth root of both sides, we get $x = 3$ (since $3^4 = 81$).

Answer:

s:

• For $\log_{7} 3 - \log_{7} 4 = \log_{7} \square$: $\frac{3}{4}$
• For $\log_{2} \square + \log_{2} 3 = \log_{2} 15$: $5$
• For $\log_{7} 81 = 4\log_{7} \square$: $3$