To solve the equation [tex]\(\log x = \log 3 + 2 \log 2 - \frac{3}{4} \log 16\)[/tex], we'll use the properties of logarithms to simplify and solve for [tex]\(x\)[/tex].
1. Rewrite the equation with logarithmic properties:
The given equation is:
[tex]\[
\log x = \log 3 + 2 \log 2 - \frac{3}{4} \log 16
\][/tex]
2. Simplify the logarithmic terms:
We can use the property of logarithms that states [tex]\(n \log a = \log(a^n)\)[/tex]. This allows us to rewrite [tex]\(2 \log 2\)[/tex] and [tex]\(\frac{3}{4} \log 16\)[/tex]:
- Simplify [tex]\(2 \log 2\)[/tex]:
[tex]\[
2 \log 2 = \log(2^2) = \log 4
\][/tex]
Given the pre-calculated value, [tex]\(\log 4 = 0.6020599913279624\)[/tex].
- Simplify [tex]\(\frac{3}{4} \log 16\)[/tex]:
[tex]\[
\frac{3}{4} \log 16 = \log(16^{\frac{3}{4}})
\][/tex]
First, simplify [tex]\(16^{\frac{3}{4}}\)[/tex]:
[tex]\[
16 = 2^4 \quad \text{so} \quad 16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^{4 \cdot \frac{3}{4}} = 2^3 = 8
\][/tex]
Thus, [tex]\(\frac{3}{4} \log 16 = \log 8\)[/tex].
Given the pre-calculated value, [tex]\(\log 16^{\frac{3}{4}} = 0.9030899869919435\)[/tex].
- The logarithm term [tex]\(\log 3\)[/tex] is directly given as [tex]\(\log 3 = 0.47712125471966244\)[/tex].
3. Combine the logarithmic terms:
The equation can now be written as:
[tex]\[
\log x = \log 3 + \log 4 - \log 8
\][/tex]
Using the properties of logarithms [tex]\(\log a + \log b = \log(ab)\)[/tex] and [tex]\(\log a - \log b = \log \left( \frac{a}{b} \right)\)[/tex], we get:
[tex]\[
\log x = \log \left(3 \cdot 4\right) - \log 8 = \log 12 - \log 8
\][/tex]
Simplifying further using the property [tex]\(\log a - \log b = \log \left( \frac{a}{b} \right)\)[/tex]:
[tex]\[
\log x = \log \left( \frac{12}{8} \right) = \log \left( \frac{3}{2} \right)
\][/tex]
Given the pre-calculated combined value of the logarithms, [tex]\(\log x = 0.17609125905568135\)[/tex].
4. Solve for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], we convert back from the logarithmic form:
[tex]\[
x = 10^{\log x}
\][/tex]
Therefore,
[tex]\[
x = 10^{0.17609125905568135} = 1.5000000000000004
\][/tex]
Hence, the solution to the equation [tex]\(\log x = \log 3 + 2 \log 2 - \frac{3}{4} \log 16\)[/tex] is:
[tex]\[
x = 1.5
\][/tex]
