Questions: Question 1 Fill in the missing values to make the equations true. (a) log9 5 - log9 4 = log9 □ (b) log8 11 + log8 □ = log8 55 (c) 2 log5 3 = log5 □

Transcript text: Question 1 Fill in the missing values to make the equations true. (a) $\log _{9} 5-\log _{9} 4=\log _{9}$ $\square$ (b) $\log _{8} 11+\log _{8}$ $\square$ $=\log _{8} 55$ (c) $2 \log _{5} 3=\log _{5}$ $\square$

Solution

Solution Steps

To solve these logarithmic equations, we can use the properties of logarithms. Specifically, the difference of logarithms can be expressed as the logarithm of a quotient, and the sum of logarithms can be expressed as the logarithm of a product. Additionally, a coefficient in front of a logarithm can be rewritten as an exponent inside the logarithm.

(a) Use the property: $\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)$.

(b) Use the property: $\log_b a + \log_b c = \log_b (a \cdot c)$.

Step 1: Solve for $ \square $ in (a)

We start with the equation: \[ \log_{9} 5 - \log_{9} 4 = \log_{9} \square \] Using the property of logarithms, we can rewrite this as: \[ \log_{9} \left(\frac{5}{4}\right) = \log_{9} \square \] Thus, we find: \[ \square = \frac{5}{4} = 1.25 \]

Step 2: Solve for $ \square $ in (b)

Next, we consider the equation: \[ \log_{8} 11 + \log_{8} \square = \log_{8} 55 \] Using the property of logarithms, we can rewrite this as: \[ \log_{8} (11 \cdot \square) = \log_{8} 55 \] This leads to: \[ 11 \cdot \square = 55 \] Solving for $ \square $: \[ \square = \frac{55}{11} = 5.0 \]

Step 3: Solve for $ \square $ in (c)

Finally, we analyze the equation: \[ 2 \log_{5} 3 = \log_{5} \square \] Using the property of logarithms, we can rewrite this as: \[ \log_{5} (3^2) = \log_{5} \square \] Thus, we find: \[ \square = 3^2 = 9 \]