Without using tables, find the value of `4log_(10)5+5log_(10)2-(1)/(2)log_(10)4`.

Answer

To solve the expression \(4 \log_{10} 5 + 5 \log_{10} 2 - \frac{1}{2} \log_{10} 4\), we can use properties of logarithms. Let's break it down step by step.

### Step 1: Rewrite the logarithmic expressions using the power rule
Using the power rule of logarithms, which states that \(m \log_b a = \log_b (a^m)\), we can rewrite the terms:

\[ 4 \log_{10} 5 = \log_{10} (5^4) \]
\[ 5 \log_{10} 2 = \log_{10} (2^5) \]
\[ -\frac{1}{2} \log_{10} 4 = \log_{10} (4^{-\frac{1}{2}}) = \log_{10} \left(\frac{1}{\sqrt{4}}\right) = \log_{10} \left(\frac{1}{2}\right) \]

### Step 2: Combine the logarithmic expressions
Now we can combine these logarithmic expressions:

\[ 4 \log_{10} 5 + 5 \log_{10} 2 - \frac{1}{2} \log_{10} 4 = \log_{10} (5^4) + \log_{10} (2^5) + \log_{10} \left(\frac{1}{2}\right) \]

Using the property that \(\log_b a + \log_b c = \log_b (a \cdot c)\), we can combine the first two terms:

\[ = \log_{10} (5^4 \cdot 2^5) + \log_{10} \left(\frac{1}{2}\right) \]

Now, we can combine these two logarithms:

\[ = \log_{10} \left(5^4 \cdot 2^5 \cdot \frac{1}{2}\right) \]

### Step 3: Simplify the expression inside the logarithm
Now simplify the expression inside the logarithm:

\[ = \log_{10} \left(5^4 \cdot 2^{5-1}\right) = \log_{10} \left(5^4 \cdot 2^4\right) \]

### Step 4: Factor out the common exponent
Since both terms have the same exponent, we can factor it out:

\[ = \log_{10} \left((5 \cdot 2)^4\right) = \log_{10} (10^4) \]

### Step 5: Evaluate the logarithm
Using the property \(\log_b (b^a) = a\):

\[ = 4 \]

Thus, the value of the expression \(4 \log_{10} 5 + 5 \log_{10} 2 - \frac{1}{2} \log_{10} 4\) is **4**.