Question

given that 2 to power of a and 8 to power of b, express a in terms of b

Answer 1

$\textsf{\large{\underline{Correct Question}:}}$

• Given that 2ᵃ = 8ᵇ. Express a in terms of b.

$\textsf{\large{\underline{Solution}:}}$

Given that:

$\rm: \longmapsto {2}^{a} = {8}^{b}$

We can also write this equation as:

$\rm: \longmapsto {2}^{a} = {( {2}^{3} )}^{b}$

We know that:

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

Therefore, we get:

$\rm: \longmapsto {2}^{a} = {2}^{3b}$

Comparing base, we get:

$\rm: \longmapsto a = 3b$

★ Which is our required answer.

$\textsf{\large{\underline{Learn More}:}}$

Laws Of Exponents: If a, b are positive real numbers and m, n are rational numbers, then the following results hold.

$\rm 1. \: \: {a}^{m} \times {a}^{n} = {a}^{m + n}$

$\rm 2. \: \: ({a}^{m})^{n} = {a}^{mn}$

$\rm 3. \: \: \dfrac{ {a}^{m} }{ {a}^{n} } = {a}^{m - n}$

$\rm4. \: \: {a}^{m} \times {b}^{m} = {(ab)}^{m}$

$\rm5. \: \: \bigg(\dfrac{a}{b} \bigg)^{m} = \dfrac{ {a}^{m} }{ {b}^{m} }$

$\rm6. \: \: {a}^{ - n} = \dfrac{1}{ {a}^{n} }$

$\rm7. \: \: {a}^{n} = {b}^{n} \rightarrow a = b, n \neq0$

$\rm8. \: \: {a}^{m} = {a}^{n} \rightarrow m = n, a \neq 1$