Find the value of `log_(8)25`, given that `log_(10)2= 0.3010`

Answer

To find the value of \( \log_{8} 25 \) given that \( \log_{10} 2 = 0.3010 \), we can follow these steps:

### Step 1: Change of Base Formula
We can use the change of base formula for logarithms, which states:
\[ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \]
Here, we will convert \( \log_{8} 25 \) to base 10:
\[ \log_{8} 25 = \frac{\log_{10} 25}{\log_{10} 8} \]

### Step 2: Express \( \log_{10} 8 \) and \( \log_{10} 25 \)
Next, we can express \( 8 \) and \( 25 \) in terms of powers of \( 2 \) and \( 5 \):
\[ 8 = 2^3 \quad \text{and} \quad 25 = 5^2 \]
Thus, we can write:
\[ \log_{10} 8 = \log_{10} (2^3) = 3 \log_{10} 2 \]
\[ \log_{10} 25 = \log_{10} (5^2) = 2 \log_{10} 5 \]

### Step 3: Substitute Values
Now we substitute these into our change of base formula:
\[ \log_{8} 25 = \frac{2 \log_{10} 5}{3 \log_{10} 2} \]

### Step 4: Find \( \log_{10} 5 \)
We can find \( \log_{10} 5 \) using the fact that \( \log_{10} 10 = 1 \):
\[ \log_{10} 10 = \log_{10} (2 \cdot 5) = \log_{10} 2 + \log_{10} 5 \]
This gives us:
\[ 1 = \log_{10} 2 + \log_{10} 5 \implies \log_{10} 5 = 1 - \log_{10} 2 \]
Substituting \( \log_{10} 2 = 0.3010 \):
\[ \log_{10} 5 = 1 - 0.3010 = 0.6990 \]

### Step 5: Substitute \( \log_{10} 5 \) Back
Now we substitute \( \log_{10} 5 \) back into our equation:
\[ \log_{8} 25 = \frac{2 \cdot 0.6990}{3 \cdot 0.3010} \]

### Step 6: Calculate the Value
Now we can calculate:
\[ \log_{8} 25 = \frac{1.3980}{0.9030} \approx 1.54817 \]

Thus, the value of \( \log_{8} 25 \) is approximately \( 1.54817 \).

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