Plane trigonometry, by S.L. Loney.

LOGARITHMS TO DIFFERENT BASES.

157

147. To prove that
loga m = logb m x loga b.
Let log, m = x, so that ac = m.
Also let logb m = y, so that by = m.. a = by.
Hence log, (ax) = loga (bY)... x= y log, b (Art. 139).
Hence log1 m = logb m x log, b.
By the theorem of the foregoing article we can from
the logarithm of any number to a base b find its logarithm
to any other base a. It is found convenient, as will appear
in a subsequent chapter, not to calculate the logarithms to
base 10 directly, but to calculate them first to another
base and then to transform them by this theorem.
EXAMPLES. XXIII.
1. Given log 4 = 60206 and log 3= '4771213, find the logarithms of
~8, -003, -0108, and (.00018)+.
2. Given log 11-=10413927 and log 13 = 1-1139434, find the values of
(1) log 1-43, (2) log 133-1, (3) log 4/143 and (4) log /-00169.
3. What are the characteristics of the logarithms of 243-7, '0153,
2'8713, '00057, -023, /24615, and (24589)-?
4. Find the 5th root of -003, having given log 3= -4771213 and
log 312936= 5-4954243.
5. Find the value of (1) 7r, (2) (84)i and (3) (.021)~, having given
log 2=30103, log 3 = 4771213,
log 7 = 8450980, log 132057 =5'1207283,
log 588453 = 5-7697117 and log 461791 = 56644438.