Question

Solve for $x$: $3({{2}^{x}}+1)-{{2}^{x+2}}+5=0$

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Hint: Take the equation $3({{2}^{x}}+1)-{{2}^{x+2}}+5=0$. Simplify the equation. Also, apply the product rule also called as Laws of indices. You will get the answer.

You must have come across the expression ${{3}^{2}}$. Here $3$ is the base and $2$ is the exponent. Exponents are also called powers or indices. The exponent of a number tells how many times to use the number in a multiplication. Let us study the laws of the exponent. It is very important to understand how the laws of exponents' laws are formulated.
Product law: According to the product law of exponents when multiplying two numbers that have the same base then we can add the exponents.
${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
Where a, m, and n all are natural numbers. Here the base should be the same in both quantities.
Quotient Law: According to the quotient law of exponents, we can divide two numbers with the same base by subtracting the exponents. In order to divide two exponents that have the same base, subtract the power in the denominator from the power in the numerator.
$\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
Power Law: According to the power law of exponents, if a there is a number raised a power to a power, just multiply the exponents.
${{({{a}^{m}})}^{n}}={{a}^{mn}}$
Exponential form refers to a numeric form that involves exponents. One method to write such a number is by identifying that each position is representing a power (exponent) of $10$. Thus, you can initially break it up into different pieces. Exponents are also called Powers or Indices.
In the question we have to find the value of $x$.
We have been given the equation,
$3({{2}^{x}}+1)-{{2}^{x+2}}+5=0$
$\Rightarrow$ $3\times {{2}^{x}}+3-{{2}^{x+2}}+5=0$
Simplifying in a simple manner we get,
$\Rightarrow$ $3\times {{2}^{x}}-{{2}^{x+2}}+8=0$
\begin{align} & 3\times {{2}^{x}}-{{2}^{x}}{{2}^{2}}=-8 \\ & 3\times {{2}^{x}}-{{2}^{x}}4=-8 \\ \end{align} ……………… (Using product law)
\begin{align} & 3\times {{2}^{x}}-{{2}^{x}}4=-8 \\ & -{{2}^{x}}=-8 \\ \end{align}
$\Rightarrow$ ${{2}^{x}}=8$
We know that, ${{2}^{3}}=8$.
So substituting in above equation we get,
${{2}^{x}}={{2}^{3}}$
$x=3$
Therefore, we get the value of $x$ is $3$.

Note: Read the question carefully. Don’t confuse yourself. You should be clear with the concept of law of exponents. Also, while simplifying do not miss any term. Kindly avoid the mistakes of signs. Solve the sum step by step so that nothing will be missed.