Given:
4log1/2^w (2log1/2^u-3log1/2^v)
Req'd:
Single logarithm = ?
Sol'n:
First remove the parenthesis,
4 log 1/2 (w) + 2 log 1/2 (u) - 3 log 1/2 (v)
Simplify each term,
Simplify the 4 log 1/2 (w) by moving the constant 4 inside the logarithm;
Simplify the 2 log 1/2 (u) by moving the constant 2 inside the logarithm;
Simplify the -3 log 1/2 (v) by moving the constant -3 inside the logarithm:
log 1/2 (w^4) + 2 log 1/2 (u) - 3 log 1/2 (v)
log 1/2 (w^4) + log 1/2 (u^2) - log 1/2 (v^3)
We have to use the product property of logarithms which is log of b (x) + log of b (y) = log of b (xy):
Thus,
Log of 1/2 (w^4 u^2) - log of 1/2 (v^3)
then use the quotient property of logarithms which is log of b (x) - log of b (y) = log of b (x/y)
Therefore,
log of 1/2 (w^4 u^2 / v^3)
and for the final step and answer, reorder or rearrange w^4 and u^2:
log of 1/2 (u^2 w^4 / v^3)
Answer:
the value of x ? answer : 4
first off, is noteworthy that's the graph of an exponential function, thus the function will be along the lines of g(x) = abˣ , now, what's "a" and "b" values?
well, let's take a peek when x = 0 and x = 1.
![\bf g(x) = ab^x \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x = 0\\ y = 1 \end{cases}\implies 1=ab^0\implies 1=a(1)\implies \boxed{1=a} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x = 1\\ y = 4 \end{cases}\implies 4 = ab^1\implies 4=1b^1\implies \boxed{4=b} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill g(x) = 4^x\qquad \qquad \qquad \begin{array}{|c|c|ll} \cline{1-2} x&y\\ \cline{1-2} -2&\frac{1}{4^2}\to \frac{1}{16}\\ -1&\frac{1}{4}\\ 0&1\\ 1&4\\ 2&16\\ \cline{1-2} \end{array}~\hfill](https://tex.z-dn.net/?f=%5Cbf%20g%28x%29%20%3D%20ab%5Ex%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20%5Cbegin%7Bcases%7D%20x%20%3D%200%5C%5C%20y%20%3D%201%20%5Cend%7Bcases%7D%5Cimplies%201%3Dab%5E0%5Cimplies%201%3Da%281%29%5Cimplies%20%5Cboxed%7B1%3Da%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20%5Cbegin%7Bcases%7D%20x%20%3D%201%5C%5C%20y%20%3D%204%20%5Cend%7Bcases%7D%5Cimplies%204%20%3D%20ab%5E1%5Cimplies%204%3D1b%5E1%5Cimplies%20%5Cboxed%7B4%3Db%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20~%5Chfill%20g%28x%29%20%3D%204%5Ex%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cbegin%7Barray%7D%7B%7Cc%7Cc%7Cll%7D%20%5Ccline%7B1-2%7D%20x%26y%5C%5C%20%5Ccline%7B1-2%7D%20-2%26%5Cfrac%7B1%7D%7B4%5E2%7D%5Cto%20%5Cfrac%7B1%7D%7B16%7D%5C%5C%20-1%26%5Cfrac%7B1%7D%7B4%7D%5C%5C%200%261%5C%5C%201%264%5C%5C%202%2616%5C%5C%20%5Ccline%7B1-2%7D%20%5Cend%7Barray%7D~%5Chfill)
