Question

Solve the given inequality. If necessary, round to four decimal places.

13^4a < 19

Answer 1

a < 0.2870

Explanation:

We are given the following inequality which we are to solve, rounding it to four decimal places:

`1 3 ^ { 4 a } < 1 9`

To solve this, we will apply the following exponent rule:

`a = b ^ { l o g _ b ( a ) }`

`19=13^{log_(13)(19)}`

Changing it back to an inequality:

`13^(4a)<13^{log_(13)(19)}`

If
`a > 1` then
`a^(f(x))<a^(g(x))` is equivalent to
`f(x)}< g(x)`.

Here,
`a=13`,
`f(x)=4a` and
`g(x)= log_(13)(19)`.

`4a<log_(13)(19)`

`a<(log_(13)(19))/(4)`

a < 0.2870

Answer 2

The solution of the inequality is a < 0.2870

Explanation:

* Lets talk about the exponential function

- the exponential function is f(x) = ab^x , where b is a constant and x

is a variable

- To solve this equation use ㏒ or ㏑

- The important rule ㏒(a^n) = n ㏒(a) OR ㏑(a^n) = n ㏑(a)

* Lets solve the problem

∵ 13^4a < 19

- To solve this inequality insert ㏑ in both sides of inequality

∴ ㏑(13^4a) < ㏑(19)

∵ ㏑(a^n) = n ㏑(a)

∴ 4a ㏑(13) < ㏑(19)

- Divide both sides by ㏑(13)

∴ 4a < ㏑(19)/㏑(13)

- To find the value of a divide both sides by 4

∴ a < [㏑(19)/㏑(13)] ÷ 4

∴ a < 0.2870

* The solution of the inequality is a < 0.2870