Questions: Calculate. [ (log ^2 8-log ^2 125)(log 0.4 4+2 log 0.4 5) ]

Transcript text: Calculate. \[ \left(\log ^{2} 8-\log ^{2} 125\right)\left(\log _{0.4} 4+2 \log _{0.4} 5\right) \]

Solution

Solution Steps

Step 1: Calculate \(\log^2 8\) and \(\log^2 125\)

We first compute the logarithms: \[ \log 8 \approx 0.9031 \quad \text{and} \quad \log 125 \approx 2.0969 \] Next, we square these values: \[ \log^2 8 \approx 0.8156 \quad \text{and} \quad \log^2 125 \approx 4.3970 \] Now, we find the difference: \[ \log^2 8 - \log^2 125 \approx 0.8156 - 4.3970 \approx -3.5815 \]

Step 2: Calculate \(\log_{0.4} 4\) and \(2 \log_{0.4} 5\)

Next, we calculate: \[ \log_{0.4} 4 \approx -1.5129 \quad \text{and} \quad \log_{0.4} 5 \approx -1.7565 \] Then, we compute: \[ 2 \log_{0.4} 5 \approx 2 \times -1.7565 \approx -3.5130 \] Now, we sum these values: \[ \log_{0.4} 4 + 2 \log_{0.4} 5 \approx -1.5129 - 3.5130 \approx -5.0259 \]

Step 3: Multiply the Results

Finally, we multiply the results from Step 1 and Step 2: \[ (-3.5815) \times (-5.0259) \approx 17.9999 \]