Logarithmic Equations.pdf - Kuta Software - Free Printable

Here's the step-by-step solution for each logarithmic equation from your worksheet. All logs without a base are assumed to be base 10 (common log), unless otherwise specified.

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1) `log(5x) = log(2x + 9)`

Since the logs are equal and have the same base, their arguments must be equal:

> `5x = 2x + 9`

Subtract `2x` from both sides:

> `3x = 9`

Divide by 3:

> `x = 3`

✔ Check domain: `5x > 0 → x > 0`, and `2x+9 > 0 → x > -4.5`. `x=3` satisfies both.

Answer: `x = 3`

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2) `log(10 - 4x) = log(10 - 3x)`

Set arguments equal:

> `10 - 4x = 10 - 3x`

Subtract 10 from both sides:

> `-4x = -3x`

Add `4x` to both sides:

> `0 = x`

✔ Check domain:
- `10 - 4x > 0 → 10 > 4x → x < 2.5`
- `10 - 3x > 0 → 10 > 3x → x < 10/3 ≈ 3.33`
`x = 0` is valid.

Answer: `x = 0`

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3) `log(4p - 2) = log(-5p + 5)`

Set arguments equal:

> `4p - 2 = -5p + 5`

Add `5p` to both sides:

> `9p - 2 = 5`

Add 2:

> `9p = 7`

Divide:

> `p = 7/9`

✔ Check domain:
- `4p - 2 > 0 → 4*(7/9) - 2 = 28/9 - 18/9 = 10/9 > 0` ✔
- `-5p + 5 > 0 → -5*(7/9) + 5 = -35/9 + 45/9 = 10/9 > 0` ✔

Answer: `p = 7/9`

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4) `log(4k - 5) = log(2k - 1)`

Set arguments equal:

> `4k - 5 = 2k - 1`

Subtract `2k`:

> `2k - 5 = -1`

Add 5:

> `2k = 4`

Divide:

> `k = 2`

✔ Check domain:
- `4k - 5 = 8 - 5 = 3 > 0` ✔
- `2k - 1 = 4 - 1 = 3 > 0` ✔

Answer: `k = 2`

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5) `log(-2a + 9) = log(7 - 4a)`

Set arguments equal:

> `-2a + 9 = 7 - 4a`

Add `4a` to both sides:

> `2a + 9 = 7`

Subtract 9:

> `2a = -2`

Divide:

> `a = -1`

✔ Check domain:
- `-2a + 9 = -2*(-1) + 9 = 2 + 9 = 11 > 0` ✔
- `7 - 4a = 7 - 4*(-1) = 7 + 4 = 11 > 0` ✔

Answer: `a = -1`

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6) `2 log₇(-2r) = 0`

Divide both sides by 2:

> `log₇(-2r) = 0`

Convert to exponential form:

> `7⁰ = -2r`

> `1 = -2r`

> `r = -1/2`

✔ Check domain: Log argument must be positive:
`-2r = -2*(-1/2) = 1 > 0` ✔

Answer: `r = -1/2`

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7) `-10 + log₃(n + 3) = -10`

Add 10 to both sides:

> `log₃(n + 3) = 0`

Convert to exponential form:

> `3⁰ = n + 3`

> `1 = n + 3`

> `n = -2`

✔ Check domain: `n + 3 = 1 > 0` ✔

Answer: `n = -2`

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8) `-2 log₃(7x) = 2`

Divide both sides by -2:

> `log₃(7x) = -1`

Convert to exponential form:

> `3^(-1) = 7x`

> `1/3 = 7x`

> `x = 1/(3*7) = 1/21`

✔ Check domain: `7x = 7*(1/21) = 1/3 > 0` ✔

Answer: `x = 1/21`

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9) `log(-m + 2) = 4`

Assume base 10.

Convert to exponential form:

> `10⁴ = -m + 2`

> `10000 = -m + 2`

Subtract 2:

> `9998 = -m`

Multiply by -1:

> `m = -9998`

✔ Check domain: `-m + 2 = -(-9998) + 2 = 9998 + 2 = 10000 > 0` ✔

Answer: `m = -9998`

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10) `-6 log₃(x - 3) = -24`

Divide both sides by -6:

> `log₃(x - 3) = 4`

Convert to exponential form:

> `3⁴ = x - 3`

> `81 = x - 3`

> `x = 84`

✔ Check domain: `x - 3 = 81 > 0` ✔

Answer: `x = 84`

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11) `log₁₂(x² + 35) = log₁₂(-12x - 1)`

Set arguments equal:

> `x² + 35 = -12x - 1`

Bring all terms to one side:

> `x² + 12x + 36 = 0`

Factor:

> `(x + 6)² = 0`

> `x = -6`

✔ Check domain:
- `x² + 35 = 36 + 35 = 71 > 0` ✔
- `-12x - 1 = -12*(-6) - 1 = 72 - 1 = 71 > 0` ✔

Answer: `x = -6`

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12) `log₉(-11x + 2) = log₉(x² + 30)`

Set arguments equal:

> `-11x + 2 = x² + 30`

Bring all terms to one side:

> `0 = x² + 11x + 28`

Factor:

> `(x + 4)(x + 7) = 0`

So `x = -4` or `x = -7`

✔ Check domain for each:

- For `x = -4`:
- `-11x + 2 = 44 + 2 = 46 > 0` ✔
- `x² + 30 = 16 + 30 = 46 > 0` ✔ → Valid

- For `x = -7`:
- `-11x + 2 = 77 + 2 = 79 > 0` ✔
- `x² + 30 = 49 + 30 = 79 > 0` ✔ → Valid

Answer: `x = -4` or `x = -7`

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## ✔ Final Answers:

1) `x = 3`
2) `x = 0`
3) `p = 7/9`
4) `k = 2`
5) `a = -1`
6) `r = -1/2`
7) `n = -2`
8) `x = 1/21`
9) `m = -9998`
10) `x = 84`
11) `x = -6`
12) `x = -4` or `x = -7`

Let me know if you want to see graphs or additional explanations!