Addition using exponents

Question

Is it possible to use exponents as a form of addition? For example in $5^{1.x} = 6$ where $x$ is the numbers after the decimal place. If so, what would be the equation to find $x$?

Answer 1

So I think you are asking how do you solve $b^{a + x} = k$ for $x$ if $b,a,k$ are constants; for example how would you solve $5^{1 + x} = 6$. Is that right.

Like so:

$5^{1 + x} = 6$

$\log_5 5^{1+x}= \log_5 6$

$1 +x = \log_5 6$

$x = \log_5 6 - 1$

$x = \frac{\ln 6}{\ln 5} - 1$

$x = 1.113...-1=0.113...$

In general $b^{a + x} = k$

if $x = \log_b k - a$

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To figure out what $\log_b k$ is ... $\log_b k = \frac{\log_x k}{\log_x b} = \frac{\log_{10} k}{\log_{10} b}= \frac{\log_e k}{\log_e b}= \frac{\ln k}{\ln b}$