Evaluate: Log25-log 125 + 1/2 log 625/3 log 5
The correct answer and explanation is :
We will evaluate the given expression step by step:
[
\log 25 – \log 125 + \frac{1}{2} \log \frac{625}{3 \log 5}
]
Step 1: Express all terms in base 5
We know that:
- ( 25 = 5^2 \Rightarrow \log 25 = \log 5^2 = 2 \log 5 )
- ( 125 = 5^3 \Rightarrow \log 125 = \log 5^3 = 3 \log 5 )
- ( 625 = 5^4 \Rightarrow \log 625 = \log 5^4 = 4 \log 5 )
Step 2: Use logarithmic properties
The given expression simplifies as follows:
[
2\log 5 – 3\log 5 + \frac{1}{2} \log \frac{4\log 5}{3\log 5}
]
Step 3: Simplify each term
[
2\log 5 – 3\log 5 = -\log 5
]
Now, evaluate the fraction inside the logarithm:
[
\frac{4\log 5}{3\log 5} = \frac{4}{3}
]
Taking the logarithm,
[
\log \frac{4}{3} = \log 4 – \log 3
]
Since ( \log 4 = 2\log 2 ) and using approximate values,
[
\log 2 \approx 0.301 \Rightarrow \log 4 = 2(0.301) = 0.602
]
Also,
[
\log 3 \approx 0.477
]
So,
[
\log \frac{4}{3} = 0.602 – 0.477 = 0.125
]
Now multiply by ( \frac{1}{2} ),
[
\frac{1}{2} \times 0.125 = 0.0625
]
Step 4: Final Computation
[
-\log 5 + 0.0625
]
Approximating ( \log 5 \approx 0.699 ),
[
-0.699 + 0.0625 = -0.6365
]
Answer:
[
\mathbf{-0.6365}
]
Explanation (300 Words)
Logarithms simplify calculations involving exponents by converting multiplication into addition and division into subtraction. In this problem, we simplify each term using logarithmic identities.
First, we express 25, 125, and 625 in terms of base 5 because logarithm properties allow simplifications:
[
\log 25 = 2 \log 5, \quad \log 125 = 3 \log 5, \quad \log 625 = 4 \log 5
]
Using the logarithm subtraction rule:
[
\log a – \log b = \log \left( \frac{a}{b} \right)
]
We simplify ( 2 \log 5 – 3 \log 5 = -\log 5 ).
Next, we evaluate ( \frac{1}{2} \log \frac{625}{3 \log 5} ), which simplifies to ( \frac{1}{2} \log \frac{4}{3} ).
Approximating logarithm values:
[
\log \frac{4}{3} = 0.125
]
Multiplying by ( \frac{1}{2} ),
[
\frac{1}{2} \times 0.125 = 0.0625
]
Finally, adding all terms gives the answer:
[
-0.6365
]
Now, I’ll generate an image related to logarithmic functions.
