Question

Write $e^{{10}^{23}}$in the form ${10}^x$, for some$x.$

Short answer

The value in the form as$e^{{10}^{23}}={10}^{\left(4.343\times {10}^{22}\right)}.$

Step-by-step solution

Step: 1 Using logarithmic function:

The natural logarithm, which has $e$as its base, is the most frequent exponential function in physics and mathematics due to its straightforward features. However, logarithms may be defined in terms of any other real number, and the definition is similar to that of natural numbers. The base ten logarithms are defined as follows:

${10}^{\log \left(x\right)}=x$

Taking logarithm on both sides and using $\ln \left(a^b\right)=b\ln \left(a\right)$,we have

$\begin{array}{l}\ln \left({10}^{\log \left(x\right)}\right)=\ln \left(x\right)\\ \log \left(x\right)\ln \left(10\right)=\ln \left(x\right)\end{array}$

Converting base as exponentiation form as

$e^{{10}^{23}}={10}^x$

Step: 2 Finding value in form:

Taking logarithm on above equation,

$\begin{array}{l}{e}^{{10}^{23}}={10}^x\\ \ln \left(e^{{10}^{23}}\right)=\ln \left({10}^x\right)\end{array}$

If $\ln \left(e^a\right)=a;\ln \left(b^a\right)=a\ln \left(b\right)$so,

$\begin{array}{l}{10}^{23}=x\ln \left(10\right)\\ x=\frac{{10}^{23}}{\ln \left(10\right)}\\ x=4.343\times {10}^{22}\\ e^{{10}^{23}}={10}^{\left(4.343\times {10}^{22}\right)}.\end{array}$