nmomma83
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Solve each equation. Use one of the 4 methods you have practiced the last few days:
1. Write exponents using the same base
2. Take the log of both sides
3. Use properties of logs then write as an exponent
4. Use properties of logs then drop the log on both sides
1. -3(10)^4-x -4=-91

Respuesta :

We'll use method 1: Write exponents using the same base.

Given equation:

[ -3(10)^{4-x} - 4 = -91 ]First, let's isolate the exponential term: [ -3(10)^{4-x} = -91 + 4 ][ -3(10)^{4-x} = -87 ]Next, let's divide both sides by -3: [ (10)^{4-x} = \frac{-87}{-3} ][ (10)^{4-x} = 29 ]Now, we'll rewrite 29 as a power of 10: [ 10^{4-x} = 10^{\log_{10}29} ][ 10^{4-x} = 10^{\log_{10}29} ]Now, since the bases are the same, the exponents must be equal: [ 4-x = \log_{10}29 ]Now, we solve for x: [ x = 4 - \log_{10}29 ]So, ( x = 4 - \log_{10}29 ).

msm555

Answer:

x ≈ 2.5376

Step-by-step explanation:

To solve the equation [tex]\sf -3 \times 10^{4-x} - 4 = -91[/tex], we'll utilize the following logarithmic rule/property:

[tex]\sf a \cdot b^x = a \cdot e^{x \cdot \log(b)}[/tex]

Using this rule, we can rewrite the equation as:

[tex]\sf -3 \cdot e^{(\log(10) \cdot (4-x))} - 4 = -91[/tex]

Now, notice that [tex]\sf \ln(10) = 1[/tex] because [tex]\sf \ln(10)[/tex] is the natural logarithm of [tex]\sf e[/tex] raised to the power of [tex]\sf 1[/tex], which is simply [tex]\sf 10[/tex].

So, the equation simplifies to:

[tex]\sf -3 \cdot e^{(4-x)} - 4 = -91[/tex]

Next, we'll add [tex]\sf 4[/tex] to both sides to isolate the exponential term:

[tex]\sf -3 \cdot e^{(4-x)} = -91 + 4[/tex]

[tex]\sf -3 \cdot e^{(4-x)} = -87[/tex]

Now, divide both sides by [tex]\sf -3[/tex] to isolate the exponential term:

[tex]\sf e^{(4-x)} = \dfrac{-87}{-3}[/tex]

[tex]\sf e^{(4-x)} = 29[/tex]

Now, we take the natural logarithm of both sides to eliminate the exponential term:

[tex]\sf \log(e^{(4-x)}) = \log(29)[/tex]

[tex]\sf 4 - x = \log(29)[/tex]

Finally, solve for [tex]\sf x[/tex]:

[tex]\sf 4 - x = \log(29)[/tex]

[tex]\sf -x = \log(29) - 4[/tex]

[tex]\sf x = 4 - \log(29)[/tex]

[tex]\sf x \approx 4 - 1.4623979978989 [/tex]

[tex]\sf x \approx 2.5376020021010 [/tex]

[tex]\sf x \approx 2.5376 \textsf{(in 4 d.p.)}[/tex]

Therefore, x ≈ 2.5376.

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