Question

How to solve $ - 14 + 3{e^x} = 11$?

Answer 1

Hint: Here we have to solve the given term by using logarithm with the base as $e$. Then we use the logarithm property to split the term. Finally we use a logarithm table to get the required answer.

Complete step-by-step solution:
The given equation is:
$ - 14 + 3{e^x} = 11$
On sending $ - 14$to the opposite side of the $ = $ sign, we get:
$\Rightarrow$$3{e^x} = 11 + 14$
On adding the right-hand side, we get:
$\Rightarrow$$3{e^x} = 25$
On sending $3$ to the opposite side of the $ = $sign, we get:
$\Rightarrow$${e^x} = \dfrac{{25}}{3}$
Now since the equation can’t be simplified anymore, we will apply log to the base $e$ on both the sides.
On applying log, we get:
$\Rightarrow$$x\ln (e) = \ln \left( {\dfrac{{25}}{3}} \right)$
Now we know that $\ln (e) = 1$therefore, using this property we can write the above equation as:
$\Rightarrow$$x = \ln \left( {\dfrac{{25}}{3}} \right)$
Now we know that given any logarithmic term, $\ln \left( {\dfrac{a}{b}} \right) = \ln (a) - \ln (b)$
Therefore, we can write the equation as:
$\Rightarrow$$x = \ln (25) - \ln (3)$
Now the value of $\ln (25)$ from the logarithm table is: $3.2188$
And the value of $\ln (3)$ from the logarithm table is: $1.0986$
On substituting the values, we get:
$\Rightarrow$$x = 3.2188 - 1.0986$
On simplifying we get:
$\Rightarrow$$x = 2.1202$, which is the required answer.

The value of x is equal to 2.1202.

Note: Logarithm is used to simplify the calculations of multiplication and division by converting them into questions of addition and subtraction respectively.
After doing calculations on the logs of the numbers, it is converted to the numeric form by using antilog.
It is to be remembered that when solving questions of exponent, logarithm can be used to simplify the terms.
It is to be remembered a logarithm has a base, which is the number to which the value of the log should be raised to get the original number.
An important property of log is ${\log _a}a = 1$.
The most commonly used bases in log are $10$ and $e$ which have a value $2.713..$ which is an irrational number.