Simplify-

1) 24 = 2) 1.53 = 3) 31.5 =

Solve for x (round to TWO decimal places if you have to)

4) 2𝑥 = 8 5) 2𝑥 = 20 6) 3𝑥 = 100

How did you go about trying to find the answer

to #6 and #7?

Goals:

1) Explain the structure and the purpose of

logarithms

2) Solve equations using logarithms

…In the 11 or 12 years you were at school you

were taught the math that took over 6000 years to

develop.

The development of ‘x’

1) x+1 = 2 → adding/subtracting

2) 2x=4 → Multiplying/dividing

3) 𝑥2 = 4 → Powers/roots

4) 𝑥 = −1 → Imaginary/complex numbers

5) 2𝑥 = 5 → exponents/logarithms

…Created logarithms to make

calculating big numbers easier

(before electronic calculators)

If you need to work with a big

messy number like:

123456789.97654321

You could instead say: For what

x will 10𝑥equal the number I

want.

Since logs follow similar rules as

other operations it makes

calculating MUCH simpler.

Magnitude 9.1 earthquake of

the coast of Indonesia in

2004, created a tsunami so

powerful in sped up the spin

of the earth by a fraction of

second.

The explosion of

Krakatoa was

about 180dB.

If you were within

40 miles of the

explosion, it would

be the last sound

you would never

hear because the

energy from the

sound wave would

burst your

eardrums before

you actually heard

the sound.

Just like addition is the inverse of subtraction and multiplication is

the inverse of division,

Notes start here:

Logarithms (or logs) are the inverse of exponents.

If 𝑓 𝑥 = 2𝑥, then logarithms answers the question

for what x will the following be true:

𝑥 = 2𝑓 𝑥

𝑓 𝑥 = 2𝑥

𝑓−1 𝑥 = 𝑙𝑜𝑔2𝑥

If 𝑏𝑥 = 𝑎 then 𝑙𝑜𝑔𝑏𝑎 = 𝑥

as long as b > 1, b ≠ 0

Exponent base Log base

Write the following exponent equation in log form:

1) 52 = 25

2) 6𝑥 = 100

3) Write your own

Write the following log equations in exponential form:

1) 𝑙𝑜𝑔232 = 8 2) 𝑙𝑜𝑔4𝑥 = 20

3) 𝑙𝑜𝑔650 = 𝑥 4) Write your own

Rewrite in log form, use the change of base formula,

solve to THREE decimal places, check:

1) Rewrite in log form: 𝑙𝑜𝑔210 = 𝑥

2) Since most calculators are only able to do log base 10 and

log base e, you need to use a change of base formula:

𝑙𝑜𝑔𝑥𝑦 =𝑙𝑜𝑔𝑦

𝑙𝑜𝑔𝑥→ 𝑙𝑜𝑔210 =

𝑙𝑜𝑔10

𝑙𝑜𝑔2

3) Solve: x= 𝑙𝑜𝑔210 =𝑙𝑜𝑔10

𝑙𝑜𝑔2= 3.322

4) Check: 23.322 = 10 (close enough)

Solve and check: 5𝑥 =15,0001) Log form → 𝑙𝑜𝑔____________ = x

2) Change of base and solve → 𝑙𝑜𝑔515000 =𝑙𝑜𝑔____

𝑙𝑜𝑔____= _______

3) Check you answer: 55.975 = 15008 (close enough but if I

needed to be more accurate I can always take more decimal

places.)