What is 5ln(10)+4-5ln(5)+1 ?

The solution to 5ln(10)+4-5ln(5)+1 is 5ln(2)+5

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simplify 5ln(10)+4-5ln(5)+1

simplify

5 ln (10) + 4 - 5 ln (5) + 1

5 ln (2) + 5

Decimal

8.46573 \dots

Solution steps

5 ln (10) + 4 - 5 ln (5) + 1

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

5 ln (10) = ln (1 0^{5})

= ln (1 0^{5}) + 4 - 5 ln (5) + 1

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

5 ln (5) = ln (5^{5})

= ln (1 0^{5}) + 4 - ln (5^{5}) + 1

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (1 0^{5}) - ln (5^{5}) = ln (\frac{1 0 ^{5}}{5 ^{5}})

= ln (\frac{1 0 ^{5}}{5 ^{5}}) + 4 + 1

\frac{1 0 ^{5}}{5 ^{5}} = 32

= ln (32) + 4 + 1

Add the numbers:

4 + 1 = 5

= ln (32) + 5

ln (32) = 5 ln (2)

= 5 ln (2) + 5

ln (b c)

ln (x e^{y})

2 lo g_{a} (b) + 3 lo g_{a} (4 b)

4 lo g_{10} (a) - lo g_{10} (c)

6 lo g_{10} (x) + lo g_{10} (10) - lo g_{10} (x + 2)