Question

Find the value \[x\] for ${9^{1 + \log x}} - {3^{1 + \log x}} - 210 = 0$. Where the base of $\log $ is 3.

Answer 1

Hint: To find the value of \[x\] first of all we have to let the term ${3^{1 + \log x}}$(which is given in the expression) equal to m (you can let any other letter according to you choice). On solving the expression we will get a quadratic equation and on solving the quadratic equation, two roots will be obtained. After finding both of the roots (m), we will insert the values of m in the expression${3^{1 + \log x}} = m$. Now, to find the value of \[x\] we have to convert the roots in the logarithm form with the base 3 (as given in the question). Now, we will compare both of the sides (L.H.S. and R.H.S.) to find the value of\[x\], where $\log $ with base 3 will be eliminated.

Formula used: $\log (a \times b) = \log a + \log b$…………………….(A)

Complete step-by-step answer:
Step 1: First of all we will let the first term of the given expression m, which is: ${3^{1 + \log x}}$.
Given expression: ${9^{1 + \log x}} - {3^{1 + \log x}} - 210 = 0$……………..(1)
Let ${3^{1 + \log x}} = m$…………………(2)
Step 2: To place the value of ${3^{1 + \log x}}$ (which we let, m in the first step) for the given expression (1), we have to convert ${9^{1 + \log x}}$in form of ${3^{1 + \log x}}$
${({3^{1 + \log x}})^2} - {3^{1 + \log x}} - 210 = 0$………………..(3)
Step 3: On placing the value of ${3^{1 + \log x}}$, which we let m in the expression (3).
${m^2} - m - 210 = 0$
Now, to solve the obtained quadratic equation we have to find the factors of 210 which are (1, 2, 3, 5, and 7). Hence we will use these factors to find the roots.
${m^2} - m(15 - 14) - 210 = 0$
${m^2} - 15m + 14m - 210 = 0$
Here, we can see that m is common in $({m^2} - 15m)$and 14 is common in $(14m - 210)$. So, obtained equation is:
$m(m - 15) + 14(m - 15) = 0$
$(m - 15)(m + 14) = 0$
Step 4: Now, to find both of the roots we will solve both of the factors.
$(m + 14) = 0$
$m = - 14$(But we can’t take this root because as we know for $\log a(a > 0)$
Now, on solving the another factor,
$
  (m - 15) = 0 \\
  m = 15 \\
 $
Step 5: Now, on placing the value of m in the equation (2).
${3^{1 + \log x}} = 15$
On taking $\log $both sides, we get
$
  {\log _3}({3^{1 + {{\log }_3}x}}) = {\log _3}(15) \\
    \\
 $
Here, we can write ${\log _3}(15)$as ${\log _3}(5 \times 3)$to solve the expression.
${\log _3}({3^{1 + \log x}}) = {\log _3}(5 \times 3)$
As we know, ${\log _a}a = 1$
So, ${\log _3}{3^{1 + {{\log }_3}x}} = 1 + {\log _3}x$
$
  1 + {\log _3}x = {\log _3}(5 \times 3) \\
    \\
 $
Step 6: As we know: ${\log _3}(5 \times 3) = {\log _3}5 + {\log _3}3$using the formula ……………..(A)
\[
  1 + {\log _3}x = {\log _3}5 + {\log _3}3 \\
    \\
 \]
We know that, \[{\log _3}3 = 1\]
Hence,
$1 + {\log _3}x = {\log _3}5 + 1$
On solving the expression we will get,
${\log _3}x = {\log _3}5$
Here, $\log $ with base 3 will be eliminated so,
$x = 5$

Hence for the given expression ${9^{1 + \log x}} - {3^{1 + \log x}} - 210 = 0$ value of x is: 5, where base of $\log $ is 3.

Note: As we let ${3^{1 + \log x}}$ as m so, we have to convert ${9^{1 + \log x}}$ in the form of ${3^{1 + \log x}}$ to replace it with ${3^{1 + \log x}}$
As we obtained a quadratic equation so there will be only two roots.
We can’t take the negative root for the logarithmic expression because as we know, $\log a(a > 0)$.
Make sure to convert all the terms of logarithm expression in the form of $\log $ with base 3.