Log function

1.
(a)
10ˣ = 1/8
10ˣ = 8⁻¹
x = log(8⁻¹)
x = ‒log(8)
x ≈ ‒0.9031

(b)
log(y) = ‒2.6
y = 10⁻²˙⁶
y = 0.002512


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2.
(a)
log(6) ‒ log(3/5)
= log[6 ÷ (3/5)]
= log[6 × (5/3)]
= log(10)
= 1

(b)
log(40) + 2log(5)
= log(40) + log(5²)
= log(40 × 25)
= log(1000)
= log(10³)
= 3log(10)
= 3

(c)
log(√7) / log(7¹ˊ²)
= log(√7) / log(√7)
= 1

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3.
5log(x³) / [6log(x) ‒ log(x¹ˊ³)]
= 15log(x) / [6log(x) ‒ (1/3)log(x)]
= 15log(x) / [6 ‒ (1/3)]log(x)
= 15 / (17/3)
= 15 × (3/17)
= 45/17


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4.
[‒2log(27) + 2.5log(9)] / [(1/2)log18 ‒ log(1/√50) ‒ 1]
= [‒log(27²) + log(3²)²˙⁵] /[log(√18) ‒log(√50)⁻¹ ‒ log(10)]
= [log(3⁵) ‒ log(27²)] /[log(√18) +log(√50) ‒ log(10)]
= log(243 / 729) / log(√18 × √50 / 10)
= log(1/3) / log(3)
= log(3)⁻¹ /log(3)
= ‒log(3) / log(3)
= ‒1

1a
10^x = 1/8
x=log (1/8)
x=-0.903

1b
logy=-2.6
y=10^(-2.6)
y=0.00251

2a
log6-log(3/5)
=log 6- log 3 +log 5
=log 10
=1

2b
log40+2log5
=log40+log25
=log1000
=3

2c
(log√7) / [log7^(1/2)]
=[log7^(1/2)]/[log7^(1/2)
=1

3
5logx³ / [6logx-logx^(1/3)]
=15logx/[(17/3)logx]
=45/17

4
(-2log27 + 2.5log9) / [(1/2)log18 - log(1/√50) - 1]
=(-6log3+5log3)+[0.5log(2*3*3)+0.5log50-1]
=(-log3)/(0.5log2+log3+0.5log2+log5-1)
=(-log3)/(log2+log3+log5-1)
=(-log3)/(log3)
=-1