What is log_{10}(4^{log_{4}(100)}) ?

The solution to log_{10}(4^{log_{4}(100)}) is 2

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simplify log_{10}(4^{log_{4}(100)})

simplify

lo g_{10} (4^{l o g_{4} (100)})

2

Solution steps

lo g_{10} (4^{l o g_{4} (100)})

lo g_{4} (100) = 2 lo g_{4} (10)

= lo g_{10} (4^{2 l o g_{4} (10)})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x), x > 0

lo g_{10} (4^{2 l o g_{4} (10)}) = lo g_{4} (100) lo g_{10} (4)

= lo g_{4} (100) lo g_{10} (4)

Apply log rule:

lo g_{a} (b) = \frac{lo g _{c} ( b )}{lo g _{c} ( a )}

lo g_{4} (100) = \frac{ln ( 100 )}{ln ( 4 )}

= \frac{ln ( 100 )}{ln ( 4 )} lo g_{10} (4)

Apply log rule:

lo g_{a} (b) = \frac{lo g _{c} ( b )}{lo g _{c} ( a )}

lo g_{10} (4) = \frac{ln ( 4 )}{ln ( 10 )}

= \frac{ln ( 100 )}{ln ( 4 )} \cdot \frac{ln ( 4 )}{ln ( 10 )}

Cross-cancel common factor:

ln (4)

= \frac{ln ( 100 )}{ln ( 10 )}

ln (100) = 2 ln (10)

= \frac{2 ln ( 10 )}{ln ( 10 )}

Cancel the common factor:

ln (10)

= 2

x^{2} (x + 2)

(\frac{1}{2} a - \frac{1}{3} b) (\frac{1}{3} a + \frac{1}{2} b)

(2 m + 1)^{2}

(2 x - 5)^{2}

100 - 36

What is log_{10}(4^{log_{4}(100)}) ?
The solution to log_{10}(4^{log_{4}(100)}) is 2