Question

SHL.
Choose the correct option.
If log(2.3) = 0.3617, then log(23) + log(0.23) + log(0.023) = .​

Answer 1

$\large\underline{\sf{Given- }}$

$\rm :\longmapsto\: log(2.3) = 0.3617$

$\large\underline{\sf{To\:Find - }}$

$\rm :\longmapsto\: log(23) + log(0.23) + log(0.023)$

$\begin{gathered}\Large{\sf{{\underline{Formula \: Used - }}}}  \end{gathered}$

$\rm :\longmapsto\: log(xy) = log(x) + log(y)$

$\rm :\longmapsto\: log( {x}^{y} ) = y log(x)$

$\rm :\longmapsto\: log(10) = 1$

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\: log(2.3) = 0.3617$

Consider,

$\rm :\longmapsto\: log(23)$

$\rm \: = \: \: log(2.3 \times 10)$

$\rm \: = \: \: log(2.3) + log(10)$

$\rm \: = \: \:0.3617 + 1$

$\rm \: = \: \:1.3617$

$\bf\implies \: log(23) = \: \:1.3617$

Now, Consider

$\rm :\longmapsto\: log(0.23)$

$\rm \: = \: \: log(2.3 \times {10}^{ - 1} )$

$\rm \: = \: \: log(2.3) + log( {10}^{ - 1} )$

$\rm \: = \: \: log(2.3) - log( {10})$

$\rm \: = \: \:0.3617 - 1$

$\rm \: = \: \: - 0.6383$

$\bf\implies \: log(0.23) = \: \: - 0.6383$

Now, Consider

$\rm :\longmapsto\: log(0.023)$

$\rm \: = \: \: log(2.3 \times {10}^{ - 2} )$

$\rm \: = \: \: log(2.3) + log( {10}^{ - 2} )$

$\rm \: = \: \: log(2.3) - 2 log( {10})$

$\rm \: = \: \:0.3617 - 2$

$\rm \: = \: \: - 1.6383$

$\bf\implies \: log(0.023) = \: \: - 1.6383$

Now, Consider

$\rm :\longmapsto\: log(23) + log(0.23) + log(0.023)$

$\rm \: = \: \:1.3617 - 0.6383 - 1.6383$

$\rm \: = \: - \:0.9149$

Hence,

$\bf\implies \:\: log(23) + log(0.23) + log(0.023) = - 0.9149$

Additional Information :-

$\rm :\longmapsto\: log_{x}(x) = 1$

$\rm :\longmapsto\: log_{x}( {x}^{y} ) = y$

$\rm :\longmapsto\: log_{ {x}^{z} }( {x}^{y} ) = \dfrac{y}{z}$

$\rm :\longmapsto\: {e}^{logx} = x$

$\rm :\longmapsto\: {e}^{ylogx} = {x}^{y}$

$\rm :\longmapsto\: {a}^{ log_{a}(x) } = x$

$\rm :\longmapsto\: {a}^{ ylog_{a}(x) } = {x}^{y}$

$\rm :\longmapsto\: log(1) = 0$

$\rm :\longmapsto\: log_{x}(y) = \dfrac{logy}{logx}$

Answer 2

Given that:

• log(2.3) = 0.3617

To Find:

• log(23) + log(0.23) + log(0.023) = ?

We know that:

• log(10) = 1

Finding the value of,

$\dashrightarrow$log(23) = log{(23/10) × 10}

$\dashrightarrow$log(23) = log(2.3) + log(10)

$\dashrightarrow$log(23) = 0.3617 + 1

$\dashrightarrow$log(23) = 1.3617

$\dashrightarrow$log(0.23) = log(2.3/10)

$\dashrightarrow$log(0.23) = log(2.3) - log(10)

$\dashrightarrow$log(0.23) = 0.3617 - 1

$\dashrightarrow$log(0.23) = - 0.6383

$\dashrightarrow$log(0.023) = log(2.3/100)

$\dashrightarrow$log(0.023) = log(2.3) - log(100)

$\dashrightarrow$log(0.023) = log(2.3) - log(10²)

$\dashrightarrow$log(0.023) = log(2.3) - 2log(10)

$\dashrightarrow$og(0.023) = 0.3617 - 2(1)

$\dashrightarrow$log(0.023) = 0.3617 - 2

$\dashrightarrow$log(0.023) = - 1.6383

Now we have:

$\dashrightarrow$log(23) = 1.3617

$\dashrightarrow$log(0.23) = - 0.6383

$\dashrightarrow$log(0.023) = - 1.6383

Finding,

$\dashrightarrow$log(23) + log(0.23) + log(0.023)

$\dashrightarrow$1.3617 + (- 0.6383) + (- 1.6383)

$\dashrightarrow$1.3617 - 0.6383 - 1.6383

$\dashrightarrow$ - 0.9149

Hence,

• log(23) + log(0.23) + log(0.023) = - 0.9149