Will someone help me with logarithm homework?

Question

4.Solve e^(p)(1/e) = 21 for p.

5. Solve 5^(x+1) - 5^x = 8 for x.

6. Solve (e^(x) - 2)^2 - e^(2x) = 2 for x.

7. Given In y^(4) + In(3y^(5)) - In2 = 20, without using a calculator, find the value of y.

8. Solve 10^(4logx) = 16 for x.

Answer 1

4.

e^p(1/e) = 21

Take logarithm to each side

loge^p(1/e) = log21

Recall : From logarithm formula logab = loga + logb

loge^p + log1/e = log(3)(7)

Recall :: From logarithm formulas  logxⁿ = nlogx and loga/b = loga - logb

p(loge) + log1 - loge =log3 + log7

Substitute log1 = 0, log3 = 0.4771  and log7 = 0.8450

p(loge) + 0 - loge = 0.4771 + 0.8450

p(loge) - loge = 1.3221

loge(p - 1) = 1.3221

Divide each side by loge

p - 1 = 1.3221/loge

Substitute loge = 0.4343

p - 1 =1.3221/0.4343

p - 1 = 3.044

Add 1 to each side

p - 1 + 1 = 3.044 + 1

p + 0 = 4.044

p = 4.044(approximately).

 

Answer 2

5.

5^x+1 - 5^x  = 8

Recall : a^x+y = (a^x)(a^y)

(5^x )(5¹) - 5^x = 8

Recall :Distributive property ab - ac =a(b - c)

5^x(5¹ - 1) = 8

5^x(5 - 1) = 8

5^x (4) = 8

Divide each side by 4

5^x (4) / (4) = 8 / 4

5^x = 2

Take logarithm to each side

log5^x = log2

Recall : logarithms formula loga^x = xloga

xlog5 = log2

Substitute log5 = 0.699 and log2 = 0.3010 then

x(0.699) = 0.3010

Divide each side by 0.699

x = 0.3010 / 0.699

x = 0.4306 (approximately).

Answer 3

6.

(e^x - 2)^2 - e^2x = 2

Recall : (a - b)² = a² - 2ab + b²

e^2x  - 4e^x + 4 - e^2x = 2

e^2x - e^2x  -4 e^x + 4 = 2

0 -4e^x+ 4 = 2

Take out common term 4

4 (-e^x + 1) = 2

Divide each side by 4

1 - e^x = 1/ 2

Add e^x to each side

1 - e^x + e^x = 1/2 + e^x

1 - 0 = e^x + 1/2

1 = e^x+ 1/2

Subtract 1/2 to each side

1 - 1/2 =e^x + 1/2 - 1/2

1/2 = e^x + 0

e^x = 1/2

Take logarithm to each side

loge^x = log1/2

Recall :loga^x = xloga and loga/b = loga - logb

Substitute loge^x = xloge and log1/2 = log1 - log2

xloge = log1 - log2

But  log1 = 0

xloge = 0 - log2

xloge = -log2

Substitue loge = 0.4343 and log2 =0.3010

x(0.4343) = -0.3010

Divide each side by 0.4343

x = -0.3010 / 0.4343

Simplify

There fore x = -0.6931 (approximately).

Answer 4

8.

10^4logx = 16

Take logarithm to each side

log10^4logx = log16

4logxlog10 = log16

Substitute log10 = 1

4logx = log16

logx⁴ = log2⁴

Take out logarithm to each side

x⁴ = 2⁴

Take 1/4 th power to each side

x = 2.

Answer 5

7) The logarthimic equation

Step - 1)

From the product rule

From the quotient rule

Step - 2)

Since we cannot take the log of a negative number, We must restrict the domain (In this case value of y ) so that

Step - 3)

Convert the original equation to an exponential equation with base e  and exponent 20.

Apply 9 th root on each side.

Solution .