Question

How do you solve ${3^x} = 729?$

Answer 1

Hint : This question describes the operation of addition/ multiplication/ division. We need to know basic logarithmic formulae involved with exponents. Also, we need to know how to convert the logarithmic addition and subtraction into logarithmic multiplication and division respectively. In this type of question, we need to find the value of $x$ from the given equation.

Complete step-by-step answer :
The given question is shown below,
${3^x} = 729 \to \left( 1 \right)$
We have to find the value $x$ from the above equation. To make easy calculation we take $\log$ on both sides of the equation $\left( 1 \right)$ , we get
$\log {3^x} = \log 729 \to \left( 2 \right)$
We know that,
$\log {a^b} = b\log a \to \left( 3 \right)$
By using the equation $\left( 3 \right)$ in the equation $\left( 2 \right)$ , we get
$\left( 2 \right) \to \log {3^x} = \log 729$
$x\log 3 = \log 729$
Let’s move the term $\log 3$ from the left side to the right side of the above equation, we get
$x = \dfrac{{\log 729}}{{\log 3}} \to \left( 4 \right)$
By using calculator we had to find that,

$$\log \left( {729} \right) = 2.8627 \\\ \log \left( 3 \right) = 0.477 \;$$

So, the equation $\left( 4 \right)$ becomes,
$x = \dfrac{{2.8627}}{{0.477}}$
$x = 6.00$
Let’s substitute the value of $x = 6.00$ in the equation $\left( 1 \right)$ we get

$$\left( 1 \right) \to {3^x} = 729 \\\ {3^6} = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 729 \;$$

So, the final answer is,

$$x = 6 \\\ {3^6} = 729 \;$$

So, the correct answer is “ x = 6”.

Note : Note that the denominator value is not to be equal to zero. This question involves the arithmetic operation of addition/ subtraction/ multiplication/ division with the involvement of logarithmic functions. Remember the basic formulae with the involvement of logarithmic function. The above-solved questions can also easily be solved by using a scientific calculator. Remember the cubic and square values of basic terms. Remember the logarithmic formula involved with exponent components. Also, note that $\dfrac{{\log \left( a \right)}}{{\log \left( b \right)}}$ can also be written as $\log \left( {\dfrac{a}{b}} \right)$ . In this method we use normal division for simplicity $a$ and $b$ , we would find the $\log$ value single term by using this method.