Question

How do you solve ${{4}^{2x+3}}=1$?

Answer 1

We have an exponential function. We will see the definition of an exponential function and a logarithmic function We will convert the given equation into its logarithm form using the relation between these two types of functions. Then we will use a change of base rule to simplify the obtained equation. After that we will solve the equation for the variable $x$.

Complete step by step answer:
The exponential function expresses a quantity and the number of times it is to be multiplied to itself. The quantity to be multiplied to itself is called the base and the number of times to be multiplied is called the exponent. For example, ${{a}^{x}}$ is an exponential function where $a$ is the base and $x$ is the exponent.
The logarithmic function is defined as the inverse of the exponential function. So, if we have the exponential function as $y={{a}^{x}}$ then its equivalent logarithmic function to this is given as ${{\log }_{a}}y=x$.
Using this relation between the exponential function and the logarithmic function, we will convert the given equation into its logarithmic form. The given equation is ${{4}^{2x+3}}=1$. So, its logarithmic form is as follows,
$2x+3={{\log }_{4}}1$
The change of base rule of logarithmic functions is given as
${{\log }_{b}}a=\dfrac{\log a}{\log b}$
Using this rule, we can write ${{\log }_{4}}1=\dfrac{\log 1}{\log 4}$. Now, we know that $\log 1=0$. Therefore, we get the following equation,
$2x+3=0$
Solving this equation for variable $x$, we get
$x=-\dfrac{3}{2}$

Note:
The conversion between logarithmic function and exponential function is very useful in calculations and simplifications. We should be familiar with the working of both these types of functions and their importance. There are other rules for logarithmic functions, like the quotient rule, product rule etc. These rules are useful for simplification of equations.