# Will someone help me with logarithm homework?

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.

4.

ep(1/e) = 21

Take logarithm to each side

logep(1/e) = log21

Recall : From logarithm formula logab = loga + logb

logep + 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

p - 1 + 1 = 3.044 + 1

p + 0 = 4.044

p = 4.044(approximately).

5.

5x+1 - 5x  = 8

Recall : ax+y = (ax)(ay)

(5x )(51) - 5x = 8

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

5x(51 - 1) = 8

5x(5 - 1) = 8

5x (4) = 8

Divide each side by 4

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

5x = 2

Take logarithm to each side

log5x = log2

Recall : logarithms formula logax = 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).

6.

(ex - 2)^2 - e2x = 2

Recall : (a - b)2 = a2 - 2ab + b2

e2x  - 4ex + 4 - e2x = 2

e2x - e2x  -4 ex + 4 = 2

0 -4ex+ 4 = 2

Take out common term 4

4 (-ex + 1) = 2

Divide each side by 4

1 - ex = 1/ 2

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

1 - 0 = ex + 1/2

1 = ex+ 1/2

Subtract 1/2 to each side

1 - 1/2 =ex + 1/2 - 1/2

1/2 = ex + 0

ex = 1/2

Take logarithm to each side

logex = log1/2

Recall :logax = xloga and loga/b = loga - logb

Substitute logex = 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).

8.

104logx = 16

Take logarithm to each side

log104logx = 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.

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 .