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Q41.

Expert-verifiedFound in: Page 352

Book edition
7th Edition

Author(s)
James Stewart, Lothar Redlin, Saleem Watson

Pages
948 pages

ISBN
9781337067508

**Logarithmic Equations Use the definition of the logarithmic function to find x.**

**(a) ${\mathrm{log}}_{2}\left(\frac{1}{2}\right)=x$**** (b) ${\mathrm{log}}_{10}x=-3$**

- The required value of
*x*is $-1$. - The required value of
*x*is 0.001.

The given equation is ${\mathrm{log}}_{2}\left(\frac{1}{2}\right)=x$.

We have to find the value of *x* using the definition of the logarithmic function.

We’ll use the definition of the logarithmic function ${\mathrm{log}}_{a}c=b\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{a}^{b}=c$.

Comparing ${\mathrm{log}}_{2}\left(\frac{1}{2}\right)=x$ with ${\mathrm{log}}_{a}c=b\text{\hspace{0.17em}}$ we get $a=2,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}b=x,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}c=\frac{1}{2}$.

So, the equivalent exponential form is:

${a}^{b}=c$

or, ${2}^{x}=\frac{1}{2}$

or, ${2}^{x}={2}^{-1}$

or, $x=-1$ [Equated the exponents, since the bases are same]

Hence, the required value of *x* is $-1$.

The given equation is ${\mathrm{log}}_{10}x=-3$.

We have to find the value of *x* using the definition of the logarithmic function.

We’ll use the definition of the logarithmic function ${\mathrm{log}}_{a}c=b\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{a}^{b}=c$.

Comparing ${\mathrm{log}}_{10}x=-3$ with ${\mathrm{log}}_{a}c=b$ we get $a=10,\text{\hspace{0.17em}\hspace{0.17em}}b=-3,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}c=x$.

So, the equivalent exponential form is:

${a}^{b}=c$

or, ${10}^{-3}=x$

or, $\frac{1}{1000}=x$

or, $0.001=x$

Hence, the required value of *x* is 0.001.

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