To simplify we need recall and apply the properties and laws of logarithms.
1)
\(log_{2}(32) + ln(e) + log_{10}(100)\)
\(log_{2}(32) + log_{e}(e) + log_{10}(100)\)
We need consider the base of each logarithm and express the numbers in the parentheses to each base raised to a certain index. or exponent.
That is
\({2}^{5} = 32 \\ \\ {10}^{2} = 100\)
As for the middle expression, the base and the number are equal so let us keep it for now.
Our expression now becomes,
\(log_{2}( {2}^{5} ) + log_{e}(e) + log_{10}( {10}^{2} )\)
Recall this law of logarithm,
\(log_{a}( {m}^{n} ) = n log_{a}(m)\)
\(5 log_{2}( {2}) + log_{e}(e) + 2log_{10}( {10} )\)
Recall again that,
\(log_{a}(a) ,a
e0 \: or \: 1\)
Therefore our expression becomes,
\(5 (1) + (1) + 2(1) = 5 + 1 + 2 = 8\)
2) We use change of base to solve this logarithmic equation.
\(log_{3}(x) + 4log_{9}(x) = 9\)
\(log_{3}(x) + log_{9}( {x}^{4} ) = 9\)
It will be easier and faster to change the base to 3.
Recall that,
\(\log_{x}(y)=\frac{log_{a}(y)}{log_{a}(x)}\)
We apply this law to obtain,
\(log_{3}(x) + \frac{log_{3}( {x}^{4})}{log_{3}(9) } = 9\)
\(log_{3}(x) + \frac{log_{3}( {x}^{4})}{log_{3}( {3}^{2} ) } = 9\)
\(log_{3}(x) + \frac{log_{3}( {x}^{4})}{2log_{3}( {3} ) } = 9\)
\(log_{3}(x) + \frac{log_{3}( {x}^{4})}{2(1) } = 9\)
\(log_{3}(x) + \frac{log_{3}( {x}^{4})}{2} = 9\)
multiplying through by 2 gives,
\(2log_{3}(x) + log_{3}( {x}^{4} ) = 18\)
\(log_{3}( {x}^{2} ) + log_{3}( {x}^{4} ) = 18\)
\(log_{3}( {x}^{4} \times {x}^{2} ) = 18\)
We apply the multiplication law of exponents to obtain,
\(log_{3}( {x}^{4 + 2} ) = 18\)
\(log_{3}( {x}^{6} ) = 18\)
We take antilogarithms to get,
\({x}^{6} = {3}^{18}\)
\({x}^{6} = ( {3}^{3} ) ^{6}\)
\(x = {3}^{3}\)
\(x = 27\)
Hence x is 27.
